# BiocalculusCalculus,Probability,andStatisticsfortheLifeSciences

Biocalculus Calculus, Probability, and Statistics for the Life Sciences Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. About the Cover Images The sex ratio of barred owl offspring is studied using probability theory in rcise 12.3.21. The fitness of a garter snake is a function of the degree of stripedness and the number of reversals of direction while fleeing a predator (rcise 9.1.7). The project on page 297 asks how birds can minimize power and energy by flapping their wings versus gliding. Example 13.3.7 uses hypothesis testing to determine if infection by malaria causes mice to become anemic. Color blindness is a genetically determined condition. Its inheritance in families is studied using conditional probability in Example 12.3.10. Data from Gregor Mendel’s famous genetic experiments with pea plants are used to introduce the techniques of descriptive statistics in Example 11.1.1. The energy needed by an iguana to run is a function of two variables, weight and speed (rcise 9.2.47). Our study of probability theory in Chapter 12 s the basis for predicting the inheritance of genetic diseases such as Huntington’s disease. The project on page 222 illustrates how mathematics can be used to minimize red blood cell loss during surgery. Jellyfish locomotion is modeled by a differential equation in rcise 10.1.34. The screw-worm fly was effectively eliminated using the sterile insect technique (rcise 5.6.24). The growth of a yeast population leads naturally to the study of differential equations (Section 7.1). The doubling time of a population of the bacterium G. lamblia is determined in rcise 1.4.29. Experimental data on EPO injection by athletes for perance enhancement are used in Chapter 13 to illustrate techniques of inferential statistics. Data on the wingspan of Monarch butterflies are used in Example 13.1.6 to illustrate the importance of sampling distributions in inferential statistics. The optimal foraging time for bumblebees is determined in Example 4.4.2. The vertical trajectory of zebra finches is modeled by a quadratic function (Figure 1.2.8). The size of the gray-wolf population depends on the size of the food supply and the number of competitors (rcise 9.4.21). Courtship displays by male ruby- throated hummingbirds provide an interesting example of a geometric random variable in rcise 12.4.72. The area of a cross-section of a human brain is estimated in rcise 6.Review.5. The project on page 479 determines the critical vaccination coverage required to eradicate a disease. Natural killer cells attack pathogens and are found in two states described by a pair of differential equations developed in Section 10.3. In Example 4.2.6 a junco has a choice of habitats with different seed densities and we determine the choice with the greatest energy reward. Data on the number of ectoparasites of damselflies are studied in rcise 11.1.9. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Australia • Brazil • Mexico • Singapore • United Kingdom • United States James Stewart McMaster University and University of Toronto Troy Day Queen’s University Biocalculus Calculus, Probability, and Statistics for the Life Sciences Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 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 ination on pricing, previous editions, changes to current editions, and alternate ats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Product Manager: Neha Taleja Senior Content Developer: Stacy Green Associate Content Developer: Samantha Lugtu Product Assistant: Stephanie Kreuz Media Developer: Lynh Pham Marketing Manager: Ryan Ahern Content Project Manager: Cheryll Linthicum Art Director: Vernon Boes Manufacturing Planner: Becky Cross Production Service and Composition: TECHarts Text and Photo Researcher: Lumina Datamatics Art and Copy Editor: Kathi Townes, TECHarts Illustrator: TECHarts Text and Cover Designer: Lisa Henry Compositor: Stephanie Kuhns, TECHarts Cover Images: DNA strand © iStockphoto.com/Frank Ramspott; Junco © Steffen Foerster/Shutterstock.com; Barred owl © mlorenz/ Shutterstock.com; Snake © Matt Jeppson/Shutterstock.com; Bird in flight © Targn Pleiades/Shutterstock.com; Red blood cells © DTKUTOO/Shutterstock.com; Ishihara test for color blindness © Eveleen/Shutterstock.com; Pea plant in bloom © yuris/Shutterstock.com; Iguana © Ryan Jackson; DNA poly- merase I © Leonid Andronov/Shutterstock.com; Surgery © Condor 36/Shutterstock.com; Jellyfish © Dreamframer/Shut- terstock.com; Srew-worm fly Courtesy of U.S. Department of Agriculture; Yeast cells © Knorre/Shutterstock.com; G. lamblia © Sebastian Kaulitzki/Shutterstock.com; Cyclist © enciktat/ Shutterstock.com; Monarch butterfly © Lightspring/Shut- terstock.com; Bumblebee foraging © Miroslav Halama/Shut- terstock.com; Bacteria rods © Fedorov Oleksiy/Shutterstock. com; Zebra finches © Wang LiQuiang/Shutterstock.com; Wolf © Vladimir Gramagin/Shutterstock.com; Hummingbird © Steve Byland/Shutterstock.com; Brain MRI © Allison Herreid/ Shutterstock.com; Syringes © Tatik22/Shutterstock.com; NK cells © Juan Gaertner/Shutterstock.com; Junco © Steffen Foerster/Shutterstock.com; Damselfly © Laura Nagel; Wolves © Vladimir Gramagin/Shutterstock.com; Snake © Matt Jeppson/Shutterstock.com; Background gradient © ririro/ Shutterstock.com Interior design images: Pills © silver-john/Shutterstock.com; Daphnia pulex © Lebendkulturen.de/Shutterstock.com © 2016 Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, ination networks, or ination storage and retri systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. Printed in the United States of America Print Number: 01 Print Year: 2015 For product ination and technology assistance, contact us at Cengage Learning Customer Semilog and Log-Log Plots 52 ■ Inverse Functions ■ Logarithmic Functions ■ Natural Logarithms ■ Graph and Growth of the Natural Logarithm ■ Semilog Plots ■ Log-Log Plots ProjeCt The Coding Function of DNA 69 1.6 Sequences and Difference Equations 70 ■ Recursive Sequences: Difference Equations ■ Discrete-Time Models in the Life Sciences ProjeCt Drug Resistance in Malaria 78 Review 80 Case study 1a Kill Curves and Antibiotic Effectiveness 84 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. viii CONTENTS 2 Limits 89 2.1 Limits of Sequences 90 ■ The Long-Term Behavior of a Sequence ■ Definition of a Limit ■ Limit Laws ■ Geometric Sequences ■ Recursion for Medication ■ Geometric Series ■ The Logistic Sequence in the Long Run ProjeCt Modeling the Dynamics of Viral Infections 101 2.2 Limits of Functions at Infinity 102 ■ The Monod Growth Function ■ Definition of a Limit at Infinity ■ Limits Involving Exponential Functions ■ Infinite Limits at Infinity 2.3 Limits of Functions at Finite Numbers 111 ■ Velocity Is a Limit ■ Limits: Numerical and Graphical s ■ One-Sided Limits ■ Infinite Limits 2.4 Limits: Algebraic s 125 ■ The Limit Laws ■ Additional Properties of Limits ■ Limits of Trigonometric Functions 2.5 Continuity 137 ■ Definition of a Continuous Function ■ Which Functions Are Continuous? ■ Approximating Discontinuous Functions by Continuous Ones Review 149 Case study 2a Hosts, Parasites, and Time-Travel 151 3 Derivatives 155 3.1 Derivatives and Rates of Change 156 ■ Measuring the Rate of Increase of Blood Alcohol Concentration ■ Tangent Lines ■ Derivatives ■ Rates of Change 3.2 The Derivative as a Function 168 ■ Graphing a Derivative from a Function’s Graph ■ Finding a Derivative from a Function’s ula ■ Differentiability ■ Higher Derivatives ■ What a Derivative Tells Us about a Function 3.3 Basic Differentiation ulas 181 ■ Power Functions ■ New Derivatives from Old ■ Exponential Functions ■ Sine and Cosine Functions 3.4 The Product and Quotient Rules 194 ■ The Product Rule ■ The Quotient Rule ■ Trigonometric Functions 3.5 The Chain Rule 202 ■ Combining the Chain Rule with Other Rules ■ Exponential Functions with Arbitrary Bases ■ Longer Chains ■ Implicit Differentiation ■ Related Rates ■ How To Prove the Chain Rule Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CONTENTS ix 3.6 Exponential Growth and Decay 215 ■ Population Growth ■ Radioactive Decay ■ Newton’s Law of Cooling ProjeCt: Controlling Red Blood Cell Loss During Surgery 222 3.7 Derivatives of the Logarithmic and Inverse Tangent Functions 222 ■ Differentiating Logarithmic Functions ■ Logarithmic Differentiation ■ The Number e as a Limit ■ Differentiating the Inverse Tangent Function 3.8 Linear Approximations and Taylor Polynomials 230 ■ Tangent Line Approximations ■ Newton’s ■ Taylor Polynomials ProjeCt: Harvesting Renewable Resources 239 Review 240 Case study 1b Kill Curves and Antibiotic Effectiveness 245 4 Applications of Derivatives 249 4.1 Maximum and Minimum Values 250 ■ Absolute and Local Extreme Values ■ Fermat’s Theorem ■ The Closed Interval ProjeCt: The Calculus of Rainbows 259 4.2 How Derivatives Affect the Shape of a Graph 261 ■ The Mean Value Theorem ■ Increasing and Decreasing Functions ■ Concavity ■ Graphing with Technology 4.3 L’Hospital’s Rule: Comparing Rates of Growth 274 ■ Indeterminate Quotients ■ Which Functions Grow Fastest? ■ Indeterminate Products ■ Indeterminate Differences ProjeCt: Mutation-Selection Balance in Genetic Diseases 284 4.4 Optimization Problems 285 ProjeCt: Flapping and Gliding 297 ProjeCt: The Tragedy of the Commons: An Introduction to Game Theory 298 4.5 Recursions: Equilibria and Stability 299 ■ Equilibria ■ Cobwebbing ■ Stability Criterion 4.6 Antiderivatives 306 Review 312 5 Integrals 315 5.1 Areas, Distances, and Pathogenesis 316 ■ The Area Problem ■ The Distance Problem ■ Pathogenesis 5.2 The Definite Integral 329 ■ Calculating Integrals ■ The Midpoint Rule ■ Properties of the Definite Integral Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. x CONTENTS 5.3 The Fundamental Theorem of Calculus 342 ■ uating Definite Integrals ■ Indefinite Integrals ■ The Net Change Theorem ■ The Fundamental Theorem ■ Differentiation and Integration as Inverse Processes ProjeCt: The Outbreak Size of an Infectious Disease 354 5.4 The Substitution Rule 354 ■ Substitution in Indefinite Integrals ■ Substitution in Definite Integrals ■ Symmetry 5.5 Integration by Parts 362 ■ Indefinite Integrals ■ Definite Integrals 5.6 Partial Fractions 368 5.7 Integration Using Tables and Computer Algebra Systems 371 ■ Tables of Integrals ■ Computer Algebra Systems ■ Can We Integrate All Continuous Functions? 5.8 Improper Integrals 376 Review 381 Case study 1c Kill Curves and Antibiotic Effectiveness 385 6 Applications of Integrals 387 6.1 Areas Between Curves 388 ■ Cerebral Blood Flow ProjeCt: Disease Progression and Immunity 394 ProjeCt: The Gini Index 395 6.2 Average Values 397 6.3 Further Applications to Biology 400 ■ Survival and Renewal ■ Blood Flow ■ Cardiac Output 6.4 Volumes 405 Review 412 Case study 1d Kill Curves and Antibiotic Effectiveness 414 Case study 2b Hosts, Parasites, and Time-Travel 416 7 Differential Equations 419 7.1 Modeling with Differential Equations 420 ■ Models of Population Growth ■ Classifying Differential Equations ProjeCt: Chaotic Blowflies and the Dynamics of Populations 430 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CONTENTS xi 7.2 Phase Plots, Equilibria, and Stability 431 ■ Phase Plots ■ Equilibria and Stability ■ A Mathematical Derivation of the Local Stability Criterion ProjeCt: Catastrophic Population Collapse: An Introduction to Bifurcation Theory 438 7.3 Direction Fields and Euler’s 440 ■ Direction Fields ■ Euler’s 7.4 Separable Equations 449 ProjeCt: Why Does Urea Concentration Rebound after Dialysis? 458 7.5 Systems of Differential Equations 459 ■ Parametric Curves ■ Systems of Two Autonomous Differential Equations ProjeCt: The Flight Path of Hunting Raptors 467 7.6 Phase Plane Analysis 468 ■ Equilibria ■ Qualitative Dynamics in the Phase Plane ProjeCt: Determining the Critical Vaccination Coverage 479 Review 480 Case study 2c Hosts, Parasites, and Time-Travel 484 8 Vectors and Matrix Models 487 8.1 Coordinate Systems 488 ■ Three-Dimensional Space ■ Higher-Dimensional Space 8.2 Vectors 496 ■ Combining Vectors ■ Components 8.3 The Dot Product 505 ■ Projections ProjeCt: Microarray Analysis of Genome Expression 513 ProjeCt: Vaccine Escape 514 8.4 Matrix Algebra 514 ■ Matrix Notation ■ Matrix Addition and Scalar Multiplication ■ Matrix Multiplication 8.5 Matrices and the Dynamics of Vectors 520 ■ Systems of Difference Equations: Matrix Models ■ Leslie Matrices ■ Summary 8.6 The Inverse and Determinant of a Matrix 528 ■ The Inverse of a Matrix ■ The Determinant of a Matrix ■ Solving Systems of Linear Equations ProjeCt: Cubic Splines 536 8.7 Eigenvectors and Eigenvalues 537 ■ Characterizing How Matrix Multiplication Changes Vectors ■ Eigenvectors and Eigenvalues Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. xii CONTENTS 8.8 Iterated Matrix Models 547 ■ Solving Matrix Models ■ Solutions with Complex Eigenvalues ■ Perron-Frobenius Theory ProjeCt: The Emergence of Geometric Order in Proliferating Cells 559 Review 560 9 Multivariable Calculus 565 9.1 Functions of Several Variables 566 ■ Functions of Two Variables ■ Graphs ■ Level Curves ■ Functions of Three Variables ■ Limits and Continuity 9.2 Partial Derivatives 585 ■ Interpretations of Partial Derivatives ■ Functions of More Than Two Variables ■ Higher Derivatives ■ Partial Differential Equations 9.3 Tangent Planes and Linear Approximations 596 ■ Tangent Planes ■ Linear Approximations ProjeCt: The Speedo LZR Racer 603 9.4 The Chain Rule 604 ■ Implicit Differentiation 9.5 Directional Derivatives and the Gradient Vector 610 ■ Directional Derivatives ■ The Gradient Vector ■ Maximizing the Directional Derivative 9.6 Maximum and Minimum Values 619 ■ Absolute Maximum and Minimum Values Review 628 10 Systems of Linear Differential Equations 631 10.1 Qualitative Analysis of Linear Systems 632 ■ Terminology ■ Saddles ■ Nodes ■ Spirals 10.2 Solving Systems of Linear Differential Equations 640 ■ The General Solution ■ Nullclines versus Eigenvectors ■ Saddles ■ Nodes ■ Spirals ■ Long-Term Behavior 10.3 Applications 652 ■ Metapopulations ■ Natural Killer Cells and Immunity ■ Gene Regulation ■ Transport of Environmental Pollutants ProjeCt: Pharmacokinetics of Antimicrobial Dosing 664 10.4 Systems of Nonlinear Differential Equations 665 ■ Linear and Nonlinear Differential Equations ■ Local Stability Analyses ■ Linearization ■ Examples Review 676 Case study 2d: Hosts, Parasites, and Time-Travel 679 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CONTENTS xiii 11 Descriptive Statistics 683 11.1 Numerical Descriptions of Data 684 ■ Types of Variables ■ Categorical Data ■ Numerical Data: Measures of Central Tendency ■ Numerical Data: Measures of Spread ■ Numerical Data: The Five-Number Summary ■ Outliers 11.2 Graphical Descriptions of Data