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INSTRUCTOR’S SOLUTIONS
fid
fid
MANUAL
en
en
tia
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ROGER LIPSETT
l
l
TO ACCOMPANY
CALCULUS FOR
BIOLOGY AND MEDICINE
FOURTH EDITION
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on
fid
en
tia
Claudia Neuhauser
l
University of Minnesota
Marcus L. Roper
University of California—Los Angeles
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on
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en
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100%pass
l
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C
on
on
fid
fid
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The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
l
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 © 2018 by Pearson Education, Inc.
Publishing as Pearson, 501 Boylston 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
C
permission of the publisher. Printed in the United States of America.
on
fid
ISBN-13: 978-0-13-412260-1
en
ISBN-10: 0-13-412260-7
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100%pass
l
,C
C
on
on
fid
fid
en
en
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Contents
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1 Preview and Review 5
1.1 Precalculus Skills Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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2 Discrete-Time Models, Sequences, and Difference Equations 87
2.1 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
on
2.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.3 Modeling with Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
fid
Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
en
3 Limits and Continuity 133
3.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
tia
3.3 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.4 Trigonometric Limits and the Sandwich Theorem . . . . . . . . . . . . . . . . . . . . . . . 159
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3.5 Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.6 A Formal Definition of Limits (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4 Differentiation 185
4.1 Formal Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.2 Properties of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.3 The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials . . 201
4.4 The Product and Quotient Rules, and the Derivatives of Rational and Power Functions . 214
4.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.6 Implicit Functions and Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 232
4.7 Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
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4.8 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
on
4.9 Derivatives of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
4.10 Derivatives of Inverse Functions, Logarithmic Functions, and the Inverse Tangent Function 254
4.11 Linear Approximation and Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 262
fid
Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
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2 CONTENTS
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5 Applications of Differentiation 275
5.1 Extrema and the Mean-Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
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5.2 Monotonicity and Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.3 Extrema and Inflection Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
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5.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.5 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
5.6 Graphing and Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
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5.7 Recurrence Equations: Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
5.8 Numerical Methods: The Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . 362
5.9 Modeling Biological Systems Using Differential Equations . . . . . . . . . . . . . . . . . . 369
5.10 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
6 Integration 399
6.1 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.2 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
6.3 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
7 Integration Techniques and Computational Methods 465
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7.1 The Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
on
7.2 Integration by Parts and Practicing Integration . . . . . . . . . . . . . . . . . . . . . . . . 471
7.3 Rational Functions and Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
7.4 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
fid
7.5 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
7.6 The Taylor Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
en
7.7 Tables of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Chapter 7 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
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8 Differential Equations 565
8.1 Solving Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
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8.2 Equilibria and Their Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
8.3 Differential Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
8.4 Integrating Factors and Two-Compartment Models . . . . . . . . . . . . . . . . . . . . . . 634
Chapter 8 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
9 Linear Algebra and Analytic Geometry 655
9.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
9.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
9.3 Linear Maps, Eigenvectors, and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 683
9.4 Demographic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
9.5 Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
Chapter 9 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
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10 Multivariable Calculus 733
10.1 Functions of Two or More Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 733
on
10.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
10.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
10.4 Tangent Planes, Differentiability, and Linearization . . . . . . . . . . . . . . . . . . . . . . 757
fid
10.5 The Chain Rule and Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 766
en
Copyright © 2018 Pearson Education, Inc.
tia
100%pass
l
C
on
on
INSTRUCTOR’S SOLUTIONS
fid
fid
MANUAL
en
en
tia
tia
ROGER LIPSETT
l
l
TO ACCOMPANY
CALCULUS FOR
BIOLOGY AND MEDICINE
FOURTH EDITION
C
on
fid
en
tia
Claudia Neuhauser
l
University of Minnesota
Marcus L. Roper
University of California—Los Angeles
C
on
fid
en
tia
100%pass
l
,C
C
on
on
fid
fid
en
en
tia
tia
l
l
C
on
fid
en
tia
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
l
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 © 2018 by Pearson Education, Inc.
Publishing as Pearson, 501 Boylston 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
C
permission of the publisher. Printed in the United States of America.
on
fid
ISBN-13: 978-0-13-412260-1
en
ISBN-10: 0-13-412260-7
tia
100%pass
l
,C
C
on
on
fid
fid
en
en
tia
tia
Contents
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l
1 Preview and Review 5
1.1 Precalculus Skills Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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2 Discrete-Time Models, Sequences, and Difference Equations 87
2.1 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
on
2.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.3 Modeling with Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
fid
Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
en
3 Limits and Continuity 133
3.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
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3.3 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.4 Trigonometric Limits and the Sandwich Theorem . . . . . . . . . . . . . . . . . . . . . . . 159
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3.5 Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3.6 A Formal Definition of Limits (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Chapter 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4 Differentiation 185
4.1 Formal Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.2 Properties of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.3 The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials . . 201
4.4 The Product and Quotient Rules, and the Derivatives of Rational and Power Functions . 214
4.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.6 Implicit Functions and Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 232
4.7 Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
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4.8 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
on
4.9 Derivatives of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
4.10 Derivatives of Inverse Functions, Logarithmic Functions, and the Inverse Tangent Function 254
4.11 Linear Approximation and Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 262
fid
Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
en
Copyright © 2018 Pearson Education, Inc.
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2 CONTENTS
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fid
5 Applications of Differentiation 275
5.1 Extrema and the Mean-Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
en
en
5.2 Monotonicity and Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.3 Extrema and Inflection Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
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5.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.5 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
5.6 Graphing and Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
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5.7 Recurrence Equations: Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
5.8 Numerical Methods: The Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . 362
5.9 Modeling Biological Systems Using Differential Equations . . . . . . . . . . . . . . . . . . 369
5.10 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Chapter 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
6 Integration 399
6.1 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
6.2 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
6.3 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
7 Integration Techniques and Computational Methods 465
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7.1 The Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
on
7.2 Integration by Parts and Practicing Integration . . . . . . . . . . . . . . . . . . . . . . . . 471
7.3 Rational Functions and Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
7.4 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
fid
7.5 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
7.6 The Taylor Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
en
7.7 Tables of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Chapter 7 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
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8 Differential Equations 565
8.1 Solving Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
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8.2 Equilibria and Their Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
8.3 Differential Equation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
8.4 Integrating Factors and Two-Compartment Models . . . . . . . . . . . . . . . . . . . . . . 634
Chapter 8 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
9 Linear Algebra and Analytic Geometry 655
9.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
9.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
9.3 Linear Maps, Eigenvectors, and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 683
9.4 Demographic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
9.5 Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
Chapter 9 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
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10 Multivariable Calculus 733
10.1 Functions of Two or More Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 733
on
10.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
10.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
10.4 Tangent Planes, Differentiability, and Linearization . . . . . . . . . . . . . . . . . . . . . . 757
fid
10.5 The Chain Rule and Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 766
en
Copyright © 2018 Pearson Education, Inc.
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100%pass
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