INSTRUCTOR’S SOLUTIONS
MANUAL
ROGER LIPSETT
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TO ACCOMPANY
CALCULUS FOR
BIOLOGY AND MEDICINE
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FOURTH EDITION
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Claudia Neuhauser
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University of Minnesota
Marcus L. Roper
University of California—Los Angeles
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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
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Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2018 by Pearson Education, Inc.
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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
permission of the publisher. Printed in the United States of America.
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ISBN-13: 978-0-13-412260-1
ISBN-10: 0-13-412260-7
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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
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Chapter 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2 Discrete-Time Models, Sequences, and Difference Equations 87
2.1 Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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2.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.3 Modeling with Recursion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Chapter 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
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3 Limits and Continuity 133
3.1 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.3 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
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3.4 Trigonometric Limits and the Sandwich Theorem . . . . . . . . . . . . . . . . . . . . . . . 159
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
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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
4.8 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
4.9 Derivatives of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
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4.10 Derivatives of Inverse Functions, Logarithmic Functions, and the Inverse Tangent Function 254
4.11 Linear Approximation and Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 262
Chapter 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
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, 2 CONTENTS
5 Applications of Differentiation 275
5.1 Extrema and the Mean-Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
5.2 Monotonicity and Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.3 Extrema and Inflection Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
5.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
5.5 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
5.6 Graphing and Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
5.7 Recurrence Equations: Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
5.8 Numerical Methods: The Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . 362
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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
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Chapter 6 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
7 Integration Techniques and Computational Methods 465
7.1 The Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
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7.2 Integration by Parts and Practicing Integration . . . . . . . . . . . . . . . . . . . . . . . . 471
7.3 Rational Functions and Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
7.4 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
7.5 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522
7.6 The Taylor Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
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7.7 Tables of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Chapter 7 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
8 Differential Equations 565
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8.1 Solving Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
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
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9.4 Demographic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
9.5 Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714
Chapter 9 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
10 Multivariable Calculus 733
10.1 Functions of Two or More Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 733
10.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
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10.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748
10.4 Tangent Planes, Differentiability, and Linearization . . . . . . . . . . . . . . . . . . . . . . 757
10.5 The Chain Rule and Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 766
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