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. 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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.