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Stay on current Cengage site**WebAssign** is a powerful digital solution designed by educators to help students learn maths and science, not just do homework. Instructors get the flexibility to define mastery thresholds and manage student learning aids down to the question level giving students the support they need to learn when they need it. **WebAssign** provides extensive content, instant assessment, and superior support.

New from James Stewart and Troy Day, the chief goal in BIOCALCULUS: CALCULUS FOR THE LIFE SCIENCES is to show students how calculus relates to biology, with a style that maintains rigor without being overly formal. The text motivates and illustrates 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. Particular attention has been paid to ensuring that all applications of the mathematics are genuine, and references to the primary biological literature for many of these has been provided so that students and instructors can explore the applications in greater depth. Although the 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 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.

PROLOGUE: MATHEMATICS AND BIOLOGY.

CASE STUDIES.

Case Study 1: Kill curves and antibiotic effectiveness.

Case Study 2: Hosts, parasites, and time travel.

1. Functions and Sequences.

1.1 Four Ways to Represent a Function.

1.2 A Catalog of Essential Functions.

1.3 New Functions from Old Functions.

Project: The Biomechanics of Human Movement.

1.4 Exponential Functions.

1.5 Logarithms; Semi-log and Log-log Plots.

Project: The Coding Function of DNA.

1.6 Sequences and Difference Equations.

Project: Drug Resistance in Malaria.

Review.

Case Study 1a: Kill curves and antibiotic effectiveness.

2. Limits.

2.1 Limits of Sequences.

Project: Modeling the Dynamics of Viral Infections.

2.2 Limits of Functions at Infinity.

2.3 Limits of Functions at Finite Numbers.

2.4 Limits: Algebraic Methods.

2.5 Continuity.

Review.

Case Study 2a: Hosts, parasites, and time travel.

3. Derivatives.

3.1 Derivatives and Rates of Change.

3.2 The Derivative as a Function.

3.3 Basic Differentiation Formulas.

3.4 The Product and Quotient Rules.

3.5 The Chain Rule.

3.6 Exponential Growth and Decay.

Project: Controlling Red Blood Cell Loss During Surgery.

3.7 Derivatives of the Logarithmic and Inverse Tangent Functions.

3.8 Linear Approximations and Taylor Polynomials.

Project: Harvesting Renewable Resources.

Review.

Case Study 1b: Kill curves and antibiotic effectiveness.

4. Applications of Derivatives.

4.1 Maximum and Minimum Values.

Project: The Calculus of Rainbows.

4.2 How Derivatives Affect the Shape of a Graph.

4.3 L'Hospital's Rule: Comparing Rates of Growth.

Project: Mutation-Selection Balance in Genetic Diseases.

4.4 Optimization.

Project: Flapping and Gliding.

Project: The Tragedy of the Commons: An Introduction to Game Theory.

4.5 Recursions: Equilibria and Stability.

4.6 Antiderivatives.

Review.

5. Integrals.

5.1 Areas, Distances, and Pathogenesis.

5.2 The Definite Integral.

5.3 The Fundamental Theorem of Calculus.

Project: The Outbreak Size of an Infectious Disease.

5.4 The Substitution Rule.

5.5 Integration by Parts.

5.6 Partial Fractions.

5.7 Integration Using Tables and Computer Algebra Systems.

5.8 Improper Integrals.

Project: Drug Bioavailability.

Review.

Case Study 1c: Kill curves and antibiotic effectiveness.

6. Applications of Integrals.

6.1 Areas Between Curves.

Project: Disease Progression and Immunity.

Project: The Gini Index.

6.2 Average Values.

6.3 Further Applications to Biology.

6.4 Volumes.

Review.

Case Study 1d: Kill curves and antibiotic effectiveness.

Case Study 2b: Hosts, parasites, and time travel.

7. Differential Equations.

7.1 Modeling with Differential Equations.

Project: Chaotic Blowflies and the Dynamics of Populations.

7.2 Phase Plots, Equilibria, and Stability.

Project: Catastrophic Population Collapse: An Introduction to Bifurcation Theory.

7.3 Direction Fields and Euler's Method.

7.4 Separable Equations.

Project: Why Does Urea Concentration Rebound After Dialysis?

7.5 Systems of Differential Equations.

Project: The Flight Path of Hunting Raptors.

7.6 Phase Plane Analysis.

Project: Determining the Critical Vaccination Coverage.

Review.

Case Study 2c: Hosts, parasites, and time travel.

8. Vectors and Matrix Models.

8.1 Coordinate Systems.

8.2 Vectors.

8.3 The Dot Product.

Project: Microarray Analysis of Genome Expression.

Project: Vaccine Escape.

8.4 Matrix Algebra.

8.5 Matrices and the Dynamics of Vectors.

8.6 The Inverse and Determinant of a Matrix.

Project: Cubic Splines.

8.7 Eigenvalues and Eigenvectors.

8.8 Iterated Linear Transformations.

Project: The Emergence of Geometric Order in Proliferating Cells.

Review.

9. Multivariable Calculus.

9.1 Functions of Several Variables.

9.2 Partial Derivatives.

9.3 Tangent Planes and Linear Approximations.

9.4 The Chain Rule .

9.5 Directional Derivatives and the Gradient Vector.

9.6 Maximum and Minimum Values.

Review.

10. Systems of Linear Differential Equations.

10.1 Qualitative Analysis of Linear Systems.

10.2 Solving Linear Systems of Differential Equations.

10.3 Applications.

Project: Pharmacokinetics of Antimicrobial Dosing

10.4 Systems of Nonlinear Differential Equations.

Review.

Case Study 2d: Hosts, parasites, and time travel.

Appendixes.

A. Intervals, Inequalities, and Absolute Values.

B. Coordinate Geometry.

C. Trigonometry.

D. Precise Definitions of Limits.

E. A Few Proofs.

F. Sigma Notation.

G. Complex Numbers.

H. Statistical Tables.

I. Glossary of Biological Terms.

J. Answers to Odd-Numbered Exercises.

- NEW: The spring 2020 eBook update includes the latest terminology. When discussing blood alcohol concentration, the units have been updated from mg/mL to g/dL. Values and art have been adjusted as needed. When discussing pathogenesis, the terminology “amount of infection” has been updated to “burden of infection.”
- NEW: Author lecture videos are now available with the teaching and learning resources.

- 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 accomplishment when completed. This text provides 24 projects. Drug Resistance in Malaria, for example, asks students to construct a recursion for the frequency of the gene that causes resistance to an antimalarial drug. The project Flapping and Gliding asks how birds can minimize power and energy by flapping their wings versus gliding. In The Tragedy of the Commons: An Introduction to Game Theory, two companies are exploiting the same fish population and students determine optimal fishing efforts. The project Disease Progression and Immunity is a nice application of areas between curves. Students use a model for the measles pathogenesis curve to determine which patients will be symptomatic and infectious (or noninfectious), or asymptomatic and noninfectious.
- Real-World Data- In order to enhance interest and conceptual understanding, it's important for students to see and work with real-world data in both numerical and graphical form. Accordingly, the text uses data concerning biological phenomena to introduce, motivate, and illustrate the concepts of calculus.
- 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 important way to foster conceptual understanding is through the problems that you assign. To that end the authors have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. Similarly, all the review sections begin with a Concept Check and a True-False Quiz. Other exercises test conceptual understanding through graphs or tables. Another type of exercise uses verbal description to test conceptual understanding.
- Biology Background - Biological background for each of the applications is given throughout the textbook. Additional information about how some biological phenomena were translated into the language of mathematics, along with animations and further references, is on the website www.stewartcalculus.com. Applications where such additional information is available are marked with the icon BB in the text.
- Technology - The availability of technology makes it more important to clearly understand the concepts that underlie the images on the screen. But, when properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. This textbook can be used either with or without technology and it uses two special symbols to indicate clearly when a particular type of machine is required. One 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 other is reserved for problems in which the full resources of a computer algebra system (like Maple, Mathematica, or the TI-89/92) are required.
- Tools for Enriching Calculus - TEC is a companion to the text and is intended to enrich and complement its contents. TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC modules that provide a laboratory environment 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 exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules.
- Enhanced WebAssign - 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 also includes a customizable YouBook, a Show Your Work feature, Just in Time review of precalculus prerequisites, an improved Assignment Editor, and an Answer Evaluator that accepts more mathematically equivalent answers and allows for homework grading in much the same way that an instructor grades.
- Case Studies - The text includes two case studies: Kill Curves and Antibiotic Effectiveness, and Host, Parasites, and Time Travel. These are extended real-world applications from the primary literature that are more involved than the projects and tie together multiple mathematical ideas throughout the book. An introduction to each case study is provided at the beginning of the book, and then each study recurs in various chapters as the student learns additional mathematical techniques. These can be used at the beginning of a course as motivation for learning the math, and they can then be returned to throughout the course as they recur in the textbook. Alternatively, one could wait until the end of a course and work through all components of a 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.

**James Stewart**

McMaster University

The late James Stewart received his M.S. from Stanford University and his Ph.D. from the University of Toronto. He did research at the University of London and was influenced by the famous mathematician George Polya at Stanford University. Stewart was most recently Professor of Mathematics at McMaster University, and his research field was harmonic analysis. Stewart was the author of a best-selling calculus textbook series published by Cengage Learning, including CALCULUS, CALCULUS: EARLY TRANSCENDENTALS, and CALCULUS: CONCEPTS AND CONTEXTS, as well as a series of precalculus texts.

**Troy Day**

Queens' University

Troy Day received his M.S. in biology from the University of British Columbia and his Ph.D. in mathematics from Queen's University. His first academic position was at the University of Toronto, before being recruited back to Queen's University as a Canada Research Chair in Mathematical Biology. He is currently a Professor of Mathematics and Statistics and a Professor of Biology. His research group works in areas ranging from applied mathematics to experimental biology. Troy Day is also coauthor of the widely used book A BIOLOGIST'S GUIDE TO MATHEMATICAL MODELING, published by Princeton University Press in 2007.

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