Biocalculus: Calculus, Probability, and Statistics for the Life Sciences PDF

 Biocalculus Calculus, Probability, and Statistics for the Life Scien...

 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 The doubling time of a population of the using probability theory in Exercise 12.3.21. bacterium G. lamblia is determined in Exercise 1.4.29. The fitness of a garter snake is a function Experimental data on EPO injection by of the degree of stripedness and the athletes for performance enhancement number of reversals of direction while are used in Chapter 13 to illustrate fleeing a predator (Exercise 9.1.7). techniques of inferential statistics. Data on the wingspan of Monarch The project on page 297 asks how birds butterflies are used in Example 13.1.6 can minimize power and energy by to illustrate the importance of sampling flapping their wings versus gliding. distributions in inferential statistics. Example 13.3.7 uses hypothesis testing The optimal foraging time for bumblebees to determine if infection by malaria is determined in Example 4.4.2. causes mice to become anemic. Color blindness is a genetically determined condition. Its inheritance The vertical trajectory of zebra finches is in families is studied using conditional modeled by a quadratic function (Figure 1.2.8). probability in Example 12.3.10. Data from Gregor Mendel’s famous The size of the gray-wolf population depends genetic experiments with pea plants on the size of the food supply and the are used to introduce the techniques of number of competitors (Exercise 9.4.21). descriptive statistics in Example 11.1.1. Courtship displays by male ruby- The energy needed by an iguana to throated hummingbirds provide an run is a function of two variables, interesting example of a geometric weight and speed (Exercise 9.2.47). random variable in Exercise 12.4.72. Our study of probability theory in Chapter 12 forms the basis for predicting The area of a cross-section of a human the inheritance of genetic diseases brain is estimated in Exercise 6.Review.5. such as Huntington’s disease. The project on page 222 illustrates how The project on page 479 determines mathematics can be used to minimize the critical vaccination coverage red blood cell loss during surgery. required to eradicate a disease. Natural killer cells attack pathogens and Jellyfish locomotion is modeled by a are found in two states described by a pair differential equation in Exercise 10.1.34. of differential equations developed in Section 10.3. In Example 4.2.6 a junco has a choice The screw-worm fly was effectively of habitats with different seed densities eliminated using the sterile insect and we determine the choice with technique (Exercise 5.6.24). the greatest energy reward. The growth of a yeast population leads Data on the number of ectoparasites of naturally to the study of differential damselflies are studied in Exercise 11.1.9. equations (Section 7.1). 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.  Biocalculus Calculus, Probability, and Statistics for the Life Sciences James Stewart McMaster University and University of Toronto Troy Day Queen’s University Australia Brazil Mexico Singapore United Kingdom United States 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 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. portant otice e ia content reference ithin the pro ct escription or the pro ct te t a not e availa le in the e oo 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. Biocalculus: Calculus, Probability, and Statistics © 2016 Cengage Learning for the Life Sciences WCN: 02-200-203 James Stewart, Troy Day ALL RIGHTS RESERVED. No part of this work covered by the copyright Product Manager: Neha Taleja herein may be reproduced, transmitted, stored, or used in any form or by Senior Content Developer: Stacy Green any means graphic, electronic, or mechanical, including but not limited to Associate Content Developer: Samantha Lugtu photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, Product Assistant: Stephanie Kreuz except as permitted under Section 107 or 108 of the 1976 United States Media Developer: Lynh Pham Copyright Act, without the prior written permission of the publisher. 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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.  To Dolph Schluter and Don Ludwig, for early inspiration 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 Authors JAMES STEWART received the M.S. degree from TROY DAY received the M.S. degree in biology Stanford University and the Ph.D. from the University from the University of British Columbia and the Ph.D. of Toronto. After two years as a postdoctoral fellow in mathematics from Queen’s University. His first at the University of London, he became Professor of academic position was at the University of Toronto, Mathematics at McMaster University. His research before being recruited back to Queen’s University as has been in harmonic analysis and functional analy- a Canada Research Chair in Mathematical Biology. sis. Stewart’s books include a series of high-school He is currently Professor of Mathematics and Sta- textbooks as well as a best-selling series of calculus tistics and Professor of Biology. His research group textbooks published by Cengage Learning. He is also works in areas ranging from applied mathematics coauthor, with Lothar Redlin and Saleem Watson, of a to experimental biology. Day is also coauthor of the series of college algebra and precalculus textbooks. widely used book A Biologist’s Guide to Mathematical Translations of his books include those into Spanish, Modeling, published by Princeton University Press in Portuguese, French, Italian, Korean, Chinese, Greek, 2007. Indonesian, and Japanese. A talented violinist, Stewart was concertmaster of the McMaster Symphony Orchestra for many years and played professionally in the Hamilton Philhar- monic Orchestra. He has given more than 20 talks worldwide on Mathematics and Music. Stewart was named a Fellow of the Fields Institute in 2002 and was awarded an honorary D.Sc. in 2003 by McMaster University. The library of the Fields Institute is named after him. The James Stewart Mathematics Centre was opened in October, 2003, at McMaster University. 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 Preface xv To the Student xxv Calculators, Computers, and Other Graphing Devices xxvi Diagnostic Tests xxviii Prologue: Mathematics and Biology xxxiii Case Studies in Mathematical Modeling xli CASE STUDY 1 Kill Curves and Antibiotic Effectiveness xlii CASE STUDY 2 Hosts, Parasites, and Time-Travel xlvi 1 Functions and Sequences 1 1.1 Four Ways to Represent a Function 2 Representations of Functions Piecewise Defined Functions Symmetry Periodic Functions Increasing and Decreasing Functions 1.2 A Catalog of Essential Functions 17 Linear Models Polynomials Power Functions Rational Functions Algebraic Functions Trigonometric Functions Exponential Functions Logarithmic Functions 1.3 New Functions from Old Functions 31 Transformations of Functions Combinations of Functions PROJECT The Biomechanics of Human Movement 40 1.4 Exponential Functions 41 The Growth of Malarial Parasites Exponential Functions Exponential Growth HIV Density and Exponential Decay The Number e 1.5 Logarithms; 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 vii 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 Methods One-Sided Limits Infinite Limits 2.4 Limits: Algebraic Methods 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 Formula Differentiability Higher Derivatives What a Derivative Tells Us about a Function 3.3 Basic Differentiation Formulas 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 Method 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 Method 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 Evaluating 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 Method 440 Direction Fields Euler’s Method 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 693 Displaying Categorical Data Displaying Numerical Data: Histograms Interpreting Area in Histograms The Normal Curve 11.3 Relationships between Variables 703 Two Categorical Variables Categorical and Numerical Variables Two Numerical Variables 11.4 Populations, Samples, and Inference 713 Populations and Samples Properties of Samples Types of Data Causation PROJECT: The Birth Weight Paradox 720 Review 722 12 Probability 727 12.1 Principles of Counting 728 Permutations Combinations 12.2 What Is Probability? 737 Experiments, Trials, Outcomes, and Events Probability When Outcomes Are Equally Likely Probability in General 12.3 Conditional Probability 751 Conditional Probability The Multiplication Rule and Independence The Law of Total Probability Bayes’ Rule PROJECT: Testing for Rare Diseases 766 12.4 Discrete Random Variables 767 Describing Discrete Random Variables Mean and Variance of Discrete Random Variables Bernoulli Random Variables Binomial Random Variables PROJECT: DNA Supercoiling 783 PROJECT: The Probability of an Avian Influenza Pandemic in Humans 784 12.5 Continuous Random Variables 786 Describing Continuous Random Variables Mean and Variance of Continuous Random Variables Exponential Random Variables Normal Random Variables Review 799 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. xiv CONTENTS 13 Inferential Statistics 803 13.1 The Sampling Distribution 804 Sums of Random Variables The Sampling Distribution of the Mean The Sampling Distribution of the Standard Deviation 13.2 Confidence Intervals 812 Interval Estimates Student’s t-Distribution 13.3 Hypothesis Testing 821 The Null and Alternative Hypotheses The t-Statistic The P-Value Summary 13.4 Contingency Table Analysis 829 Hypothesis Testing with Contingency Tables The Chi-Squared Test Statistic The Hypothesis Test Summary Review 835 APPENDIXES 839 A Intervals, Inequalities, and Absolute Values 840 B Coordinate Geometry 845 C Trigonometry 855 D Precise Definitions of Limits 864 E A Few Proofs 870 F Sigma Notation 874 G Complex Numbers 880 H Statistical Tables 888 GLOSSARY OF BIOLOGICAL TERMS 891 ANSWERS TO ODD-NUMBERED EXERCISES 893 BIOLOGICAL INDEX 947 INDEX 957 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.  Preface In recent years more and more colleges and universities have been introducing calculus courses specifically for students in the life sciences. This reflects a growing recognition that mathematics has become an indispensable part of any comprehensive training in the biological sciences. Our chief goal in writing this textbook is to show students how calculus relates to biology. We motivate and illustrate the topics of calculus with examples drawn from many areas of biology, including genetics, biomechanics, medicine, pharmacology, physiology, ecology, epidemiology, and evolution, to name a few. We have paid par- ticular attention to ensuring that all applications of the mathematics are genuine, and we provide references to the primary biological literature for many of these so that students and instructors can explore the applications in greater depth. We strive for a style that maintains rigor without being overly formal. Although our focus is on the interface between mathematics and the life sciences, the logical structure of the book is motivated by the mathematical material. Students will come away from a course based on this book with a sound knowledge of mathematics and an understanding of the importance of mathematical arguments. Equally important, they will also come away with a clear understanding of how these mathematical concepts and techniques are central in the life sciences, just as they are in physics, chemistry, and engineering. The book begins with a prologue entitled Mathematics and Biology detailing how the applications of mathematics to biology have proliferated over the past several decades and giving a preview of some of the ways in which calculus provides insight into biologi- cal phenomena. Alternate Versions There are two versions of this textbook. The first is entitled Biocalculus: Calculus for the Life Sciences; it focuses on calculus and some elements of linear algebra that are important in the life sciences. This is the second version, entitled Biocalculus: Calculus, Probability, and Statistics for the Life Sciences; it contains all of the content of the first version as well as three additional chapters titled Descriptive Statistics, Probability, and Inferential Statistics (see Content on page xviii). Features Real-World Data We think it’s important for students to see and work with real-world data in both numeri- cal and graphical form. Accordingly, we have used data concerning biological phenom- ena to introduce, motivate, and illustrate the concepts of calculus. Many of the examples and exercises deal with functions defined by such numerical data or graphs. See, for example, Figure 1.1.1 (electrocardiogram), Figure 1.1.23 (malarial fever), Exercise 1.1.26 (blood alcohol concentration), Table 2 in Section 1.4 (HIV density), Table 3 in Section 1.5 (species richness in bat caves), Example 3.1.7 (growth of malarial parasites), xv 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. xvi PREFACE Exercise 3.1.42 (salmon swimming speed), Exercises 4.1.7–8 (influenza pandemic), Exercise 4.2.10 (HIV prevalence), Figure 5.1.17 (measles pathogenesis), Exercise 5.1.11 (SARS incidence), Figure 6.1.8 and Example 6.1.4 (cerebral blood flow), Table 1 and Figure 1 in Section 7.1 (yeast population), Figure 8.1.14 (antigenic cartography), Exer- cises 9.1.7, 9.2.48, and Examples 9.5.5 and 9.6.6 (snake reversals and stripes). And, of course, Chapters 11 and 13 are focused entirely on the analysis of biological data. Graded Exercise Sets Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems involving applications and proofs. Conceptual Exercises One of the goals of calculus instruction is conceptual understanding, and the most impor- tant way to foster conceptual understanding is through the problems that we assign. To that end we have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first few exercises in Sections 2.3, 2.5, 3.3, 4.1, and 8.2.) Similarly, all the review sections begin with a Concept Check and a True-False Quiz. Other exercises test concep- tual understanding through graphs or tables (see Exercises 3.1.11, 5.2.41–43, 7.1.9–11, 9.1.1–2, and 9.1.26–32). Another type of exercise uses verbal description to test conceptual understanding (see Exercises 2.5.12, 3.2.50, 4.3.47, and 5.8.29). Projects One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplish- ment when completed. We have provided 24 projects in Biocalculus: Calculus for the Life Sciences and an additional four in Biocalculus: Calculus, Probability, and Statistics for the Life Sciences. Drug Resistance in Malaria (page 78), for example, asks students to construct a recursion for the frequency of the gene that causes resistance to an anti- malarial drug. The project Flapping and Gliding (page 297) asks how birds can mini- mize power and energy by flapping their wings versus gliding. In The Tragedy of the Commons: An Introduction to Game Theory (page 298), two companies are exploiting the same fish population and students determine optimal fishing efforts. The project Dis- ease Progression and Immunity (page 394) is a nice application of areas between curves. DNA Supercoiling (page 783) uses ideas from probability theory to predict how DNA is coiled and compacted into cells. We think that, even when projects are not assigned, students might well be intrigued by them when they come across them between sections. Case Studies We also provide two case studies: (1) Kill Curves and Antibiotic Effectiveness and (2) Hosts, Parasites, and Time-Travel. These are extended real-world applications from the primary literature that are more involved than the projects and that tie together mul- tiple mathematical ideas throughout the book. An introduction to each case study is pro- vided at the beginning of the book (page xli), and then each case study recurs in various chapters as the student learns additional mathematical techniques. The case studies can be used at the beginning of a course as motivation for learning the mathematics, and they can then be returned to throughout the course as they recur in the textbook. Alternatively, 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.  PREFACE xvii a case study may be assigned at the end of a course so students can work through all com- ponents of the case study in its entirety once all of the mathematical ideas are in place. Case studies might also be assigned to students as term projects. Additional case studies will be posted on the website www.stewartcalculus.com as they become available. Biology Background Although we give the biological background for each of the applications throughout the textbook, it is sometimes useful to have additional information about how the biological phenomenon was translated into the language of mathematics. In order to maintain a clear and logical flow of the mathematical ideas in the text, we have therefore included such information, along with animations, further references, and downloadable data on the website www.stewartcalculus.com. Applications for which such additional informa- tion is available are marked with the icon BB in the text. Technology The availability of technology makes it more important to clearly understand the con- cepts that underlie the images on the screen. But, when properly used, graphing calcula- tors and computers are powerful tools for discovering and understanding those concepts. (See the section Calculators, Computers, and Other Graphing Devices on page xxvi for a discussion of these and other computing devices.) These textbooks can be used either with or without technology and we use two special symbols to indicate clearly when a particular type of machine is required. The icon ; indicates an exercise that definitely requires the use of such technology, but that is not to say that it can’t be used on the other exercises as well. The symbol CAS is reserved for problems in which the full resources of a computer algebra system (like Maple, Mathematica, or the TI-89/92) are required. But technology doesn’t make pencil and paper obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where the hand or the machine is appropriate. Tools for Enriching Calculus (TEC) TEC is a companion to the text and is intended to enrich and complement its contents. (It is now accessible in Enhanced WebAssign and CengageBrain.com. Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed in collaboration with Harvey Keynes, Dan Clegg, and Hubert Hohn, TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons TEC direct students to TEC Visuals and Modules that provide a laboratory environ- ment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent explo- ration, to assigning specific exercises from those included with each Module, to creating additional exercises, labs, and projects that make use of the Visuals and Modules. Enhanced WebAssign Technology is having an impact on the way homework is assigned to students, particu- larly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. We have been working with the calculus 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. xviii PREFACE community and WebAssign to develop a robust online homework system. Up to 50% of the exercises in each section are assignable as online homework, including free response, multiple choice, and multi-part formats. The system also includes Active Examples, in which students are guided in step-by- step tutorials through text examples, with links to the textbook and to video solutions. The system features a customizable YouBook, a Show My Work feature, Just in Time review of precalculus prerequisites, an Assignment Editor, and an Answer Evaluator that accepts mathematically equivalent answers and allows for homework grading in much the same way that an instructor grades. Website The site www.stewartcalculus.com includes the following. Algebra Review Lies My Calculator and Computer Told Me History of Mathematics, with links to the better historical websites Additional Topics (complete with exercise sets): Approximate Integration: The Trapezoidal Rule and Simpson’s Rule, First-Order Linear Differential Equations, Second-Order Linear Differential Equations, Double Integrals, Infinite Series, and Fourier Series Archived Problems (drill exercises and their solutions) Challenge Problems Links, for particular topics, to outside Web resources Selected Tools for Enriching Calculus (TEC) Modules and Visuals Case Studies Biology Background material, denoted by the icon BB in the text Data sets Content Diagnostic Tests The books begin with four diagnostic tests, in Basic Algebra, Ana- lytic Geometry, Functions, and Trigonometry. Prologue This is an essay entitled Mathematics and Biology. It details how the appli- cations of mathematics to biology have proliferated over the past several decades and highlights some of the applications that will appear throughout the book. Case Studies The case studies are introduced here so that they can be used as moti- vation for learning the mathematics. Each case study then recurs at the ends of various chapters throughout the book. 1 Functions and Sequences The first three sections are a review of functions from precalculus, but in the context of biological applications. Sections 1.4 and 1.5 review exponential and logarithmic functions; the latter section includes semilog and log- log plots because of their importance in the life sciences. The final section introduces sequences at a much earlier stage than in most calculus books. Emphasis is placed on recursive sequences, that is, difference equations, allowing us to discuss discrete-time models in the biological 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.  PREFACE xix 2 Limits We begin with limits of sequences as a follow-up to their introduction in Section 1.6. We feel that the basic idea of a limit is best understood in the context of sequences. Then it makes sense to follow with the limit of a function at infinity, which we present in the setting of the Monod growth function. Then we consider limits of functions at finite numbers, first geometrically and numerically, then algebraically. (The precise definition is given in Appendix D.) Continuity is illustrated by population har- vesting and collapse. 3 Derivatives Derivatives are introduced in the context of rate of change of blood alcohol concentration and tangent lines. All the basic functions, including the exponen- tial and logarithmic functions, are differentiated here. When derivatives are computed in applied settings, students are asked to explain their meanings. 4 Applications of Derivatives The basic facts concerning extreme values and shapes of curves are deduced using the Mean Value Theorem as the starting point. In the sec- tion on l’Hospital’s Rule we use it to compare rates of growth of functions. Among the applications of optimization, we investigate foraging by bumblebees and aquatic birds. The Stability Criterion for Recursive Sequences is justified intuitively and a proof based on the Mean Value Theorem is given in Appendix E. 5 Integrals The definite integral is motivated by the area problem, the distance prob- lem, and the measles pathogenesis problem. (The area under the pathogenesis curve up to the time symptoms occur is equal to the total amount of infection needed to develop symptoms.) Emphasis is placed on explaining the meanings of integrals in various con- texts and on estimating their values from graphs and tables. There is no separate chapter on techniques of integration, but substitution and parts are covered here, as well as the simplest cases of partial fractions. 6 Applications of Integrals The Kety-Schmidt method for measuring cerebral blood flow is presented as an application of areas between curves. Other applications include the average value of a fish population, blood flow in arteries, the cardiac output of the heart, and the volume of a liver. 7 Differential Equations Modeling is the theme that unifies this introductory treat- ment of differential equations. The chapter begins by constructing a model for yeast pop- ulation size as a way to motivate the formulation of differential equations. We then show how phase plots allow us to gain considerable qualitative information about the behavior of differential equations; phase plots also provide a simple introduction to bifurcation theory. Examples range from cancer progression to individual growth, to ecology, to anesthesiology. Direction fields and Euler’s method are then studied before separable equations are solved explicitly, so that qualitative, numerical, and analytical approaches are given equal consideration. The final two sections of this chapter explore systems of two differential equations. This brief introduction is given here because it allows students to see some applications of systems of differential equations without requiring any addi- tional mathematical preparation. A more complete treatment is then given in Chapter 10. 8 Vectors and Matrix Models We start by introducing higher-dimensional coordi- nate systems and their applications in the life sciences including antigenic cartography and genome expression profiles. Vectors are then introduced, along with the dot product, and these are shown to provide insight ranging from influenza epidemiology, to cardiol- ogy, to vaccine escape, to the discovery of new biological compounds. They also provide some of the tools necessary for the treatment of multivariable calculus in Chapter 9. The remainder of this chapter is then devoted to the application of further ideas from linear algebra to biology. A brief introduction to matrix algebra is followed by a section where these ideas are used to model many different biological phenomena with the aid of matrix diagrams. The final three sections are devoted to the mathematical analysis of Copyright 2016 Ceng