## Summary

This volume teaches calculus in thebiologycontextwithoutcompromising the level of regular calculus. The material is organized in the standard way and explains how the different concepts are logically related. Each new concept is typically introduced with a biological example; the concept is then developedwithoutthe biological context and then the concept is tied into additional biological examples. This allows readers to first seewhya certain concept is important, then lets them focus on how to use the conceptswithoutgetting distracted by applications, and then, once readers feel more comfortable with the concepts, it revisits the biological applications to make sure that they canapplythe concepts. The book features exceptionally detailed, step-by-step, worked-out examples and a variety of problems, including an unusually large number of word problems.The volume begins with a preview and review and moves into discrete time models, sequences, and difference equations, limits and continuity, differentiation, applications of differentiation, integration techniques and computational methods, differential equations, linear algebra and analytic geometry, multivariable calculus, systems of differential equations and probability and statistics.For faculty and postdocs in biology departments.

## Table of Contents

** 1. Preview and Review. ** Preliminaries. Elementary Functions. Graphing. Key Terms. Review Problems.

** 2. Discrete Time Models, Sequences, and Difference Equations. ** Exponential Growth and Decay. Sequences. More Population Models. Key Terms. Review Problems.

** 3. Limits and Continuity. ** Limits. Continuity. Limits at Infinity. The Sandwich Theorem and Some Trigonometric Limits. Properties of Continuous Functions. Formal Definition of Limits. Key Terms. Review Problems.

** 4. Differentiation. ** Formal Definition of the Derivative. The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials. Product and Quotient Rule. The Chain Rule and Higher Derivatives. Derivatives of Trigonometric Functions. Derivatives of Exponential Functions. Derivatives of Inverse and Logarithmic Functions. Approximation and Local Linearity. Key Terms. Review Problems.

** 5. Applications of Differentiation. ** *Extrema and the Mean Value Theorem. Monotonicity and Concavity. Extrema, Inflection Points and Graphing. Optimization. L'Hospital's Rule. Numerical Methods—The Newton-Raphson Methods. Antiderivatives. Key Terms. Review Problems. *

** 6. Integration. ** The Definite Integral. The Fundamental Theorem of Calculus. Applications of Integration. Key Terms. Review Problems.

** 7. Integration Techniques and Computational Methods. ** The Substitution Rule. Integration by Parts. Practicing Integration and Partial Fractions. Improper Integrals. Numerical Integration. Tables of Integration. The Taylor Approximation. Key Terms. Review Problems.

** 8. Differential Equations. ** Solving Differential Equations. Equilibria and Their Stability. Systems of Autonomous Equations. Key Terms. Review Problems.

** 9. Linear Algebra and Analytic Geometry. ** Linear Systems. Matrices. Linear Maps, Eigenvectors and Eignvalues. Analytic Geometry. Key Terms. Review Problems.

** 10. Multivariable Calculus. ** Functions of Two or More Independent Variables. Limits and Continuity. Partial Derivatives. Tangent Planes, Differentiability, and Linearization. More About Derivatives. Applications. Systems of Difference Equations. Key Terms. Review Problems.

** 11. Systems of Differential Equations. ** Linear Systems—Theory. Linear Systems—Applications. Nonlinear Autonomous Systems—Theory. Nonlinear Systems—Applications. Key Terms. Review Problems.

** 12. Probability and Statistics. ** Counting. What Is Probability? Conditional Probability and Independence. Discrete Random Variables and Discrete Distributions. Continuous Distributions. Limit Theorems. Statistical Tools. Key Terms. Review Problems.

## Excerpts

When the first edition of this book appeared three years ago, I immediately started thinking about the second edition. Topics absent from the first edition were needed in a calculus book for life science majors: difference equations, extrema for functions of two variables, optimization under constraints, and expanded probability theory. I also wanted to add more biological examples, in particular in the first half of the book, and add more problems (the number of problems in many sections doubled or tripled compared with the first edition).Despite these changes, the goals of the first edition remain: To model and analyze phenomena in the life sciences using calculus. do a traditional calculus course, biology students rarely see why the material is relevant to their training. This text is written exclusively for students in the biological and medical sciences. It makes an effort to show them from the beginning how calculus can help to understand phenomena in nature.This text differs from traditional calculus texts.First,it is written in a life science context; concepts are motivated with biological examples to emphasize that calculus is an important tool in the life sciences. The second edition has many more biological examples than the first edition, particularly in the first half of the book.Second,difference equations are now extensively treated in the book. They are introduced in Chapter 2, where they are accessible to calculus students without a knowledge of calculus and provide an easier entrance to population models than differential equations. They are picked up again in Chapters 5 and 10, where they receive a more formal treatment using calculus.Third,differential equations, one of the most important modeling tools in the life sciences, are introduced early, immediately after the formal definition of derivatives in Chapter 4. Two chapters deal exclusively with differential equations and systems of differential equations; both chapters contain numerous up-to-date applications.Fourth,biological applications of differentiation and integration are integrated throughout the text.Fifth,multivariable calculus is taught in the first year, recognizing that most students in the life sciences will not take the second year of calculus and that multivariable calculus is needed to analyze systems of difference and differential equations, which students encounter later in their science courses. The chapter on multivariable calculus now has a treatment of extrema and Lagrange multipliers.This text does not teach modeling; the objective is to teach calculus. Modeling is an art that should' be taught in a separate course. However, throughout the text, students encounter mathematical models for biological phenomena. This will facilitate the transition to actual modeling and allows them to see how calculus provides useful tools for the life sciences.Examples.Each topic is motivated with biological examples. This is followed by a thorough discussion outside of the life science context to enable students to become familiar with both the meaning and the mechanics of the topic. Finally, biological examples are given to teach students how to use the material in a life science context.Examples in the text are completely worked out; steps in calculation are frequently explained in words.Problems.Calculus cannot be learned by watching someone do it. This is recognized by providing the students with both drill and word problems. Word problems are an integral part of teaching calculus in a life science context. The word problems are up to date; they are adapted from either standard biology texts or original research. Many new problems have been added to the second edition. Since this text is written for c911ege freshmen, the examples were chosen so that no formal training in biology is needed.Technology.The book takes advantage of graphing ca