Synopses & Reviews
This volume teaches calculus in the biology context without compromising 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 developed without the biological context and then the concept is tied into additional biological examples. This allows readers to first see why a certain concept is important, then lets them focus on how to use the concepts without getting distracted by applications, and then, once readers feel more comfortable with the concepts, it revisits the biological applications to make sure that they can apply the 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.
Calculus for Biology and Medicine, Third Edition, addresses the needs of readers in the biological sciences by showing them how to use calculus to analyze natural phenomena—without compromising the rigorous presentation of the mathematics. While the table of contents aligns well with a traditional calculus text, all the concepts are presented through biological and medical applications. The text provides readers with the knowledge and skills necessary to analyze and interpret mathematical models of a diverse array of phenomena in the living world. This book is suitable for a wide audience, as all examples were chosen so that no formal training in biology is needed.
About the Author
Claudia Neuhauser is Vice Chancellor for Academic Affairs and Director of the Center for Learning Innovation at the University of Minnesota Rochester (UMR). She is a Distinguished McKnight University Professor, Howard Hughes Medical Institute Professor, and Morse-Alumni Distinguished Teaching Professor. She received her diploma in Mathematics from the Universität Heidelberg (Germany), and a PhD in Mathematics from Cornell University. Before joining UMR in July 2008, she was Professor and Head in the Department of Ecology, Evolution and Behavior at the University of Minnesota—Twin Cities, and a faculty member in mathematics departments at the University of Southern California, University of Wisconsin—Madison, University of Minnesota, and University of California—Davis. Dr. Neuhauser’s research is at the interface of ecology and evolution. She investigates effects of spatial structure on community dynamics, in particular, the effect of competition on the spatial structure of competitors and the effect of symbionts on the spatial distribution of their hosts. In addition, her research in population genetics has resulted in the development of statistical tools for random samples of genes. In her role as Director of the Center for Learning Innovation at the University of Minnesota Rochester, Dr. Neuhauser is responsible for the development of the Bachelor of Science in Health Sciences. The Center promotes a learner-centered, concept-based learning environment in which ongoing assessment guides and monitors student learning and is the basis for data-driven research on learning. Dr. Neuhauser’s interest in furthering the quantitative training of biology undergraduate students has resulted in a textbook on Calculus for Biology and Medicine and a web page Numb3r5 Count! (http://bioquest.org/numberscount/). In her spare time, she enjoys riding her bike, working out in the gym, and reading history and philosophy.
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.
Formal Definition of the Derivative. The Power Rule, the Basic Rules of Differentiation, and the Derivatives of Polynomials. Product Rule 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'Hopital's Rule. Difference Equations - Stability. Numerical Methods: The Newton-Raphson Method. Antiderivatives. Key Terms. Review Problems.
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.