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Designed to help life sciences students understand the role mathematics has played in breakthroughs in epidemiology, genetics, statistics, physiology, and other biological areas, MODELING THE DYNAMCICS OF LIFE: CALCULUS AND PROBABILTY FOR LIFE SCIENTISTS, Third Edition, provides students with a thorough grounding in mathematics, the language, and 'the technology of thought' with which these developments are created and controlled. The text teaches the skills of describing a system, translating appropriate aspects into equations, and interpreting the results in terms of the original problem. The text helps unify biology by identifying dynamical principles that underlie a great diversity of biological processes. Standard topics from calculus courses are covered, with particular emphasis on those areas connected with modeling such as discrete-time dynamical systems, differential equations, and probability and statistics.
- Section 1.7 includes double-log graphs and an introduction to allometry, the study of power function relationships among biological measurements.
- Section 2.8 includes examples of implicit differentiation and related rates with both geometric and dynamical applications.
- Infinite series are introduced first in a new discussion of Taylor series in section 3.7, and then studied more formally in the context of improper integrals in section 4.7.
- Integration by partial fractions is introduced in section 4.3 and used to solve the logistic differential equation in section 5.4. Trigonometric substitutions are presented in a new series of exercises in section 4.3.
- Computing volumes of solids of revolution is introduced in section 4.6.
- Three dynamical principles that underlie diverse biological processes are woven throughout: growth, diffusion, and selection. Each theme is studied in turn with the three kinds of models.
- The final three chapters teach probability and statistics from a dynamical perspective.
- The author emphasizes the link between models of key processes and the fundamental statistical notions of likelihood, estimation, and hypothesis testing, so that students learn the principles of statistics rather than learning how to choose the correct formulas.
- Graphical and computer techniques are introduced and used throughout to support the text's focus on reasoning and interpreting models.
- Approximately 30 algorithms are clearly identified so that students can use them throughout the course.
- Several extended explorations are included to show students how to combine a set of processes into a coherent whole, and how to use models to clarify and answer specific questions.
- Each section of the text includes a wide variety of modeling problems, in addition to review problems.
- Each chapter includes at least two projects, suitable for group work, and concludes with supplementary application problems.
- The text contains more than 100 graphing calculator or computer exercises, designed to help students visualize and conceptualize key concepts.
- The first chapter includes a review of functions, units, and linear functions before beginning with the new topic of discrete-time dynamical systems.
- Partial solutions to all the odd problems are included in the back of the book, providing valuable guidance for students at no additional cost.
- The text integrates mathematical content with modeling, following the process of describing a system, translating appropriate aspects into equations, and interpreting results in terms of the original problem.
- The text introduces and develops mathematical methods to analyze three kinds of models: discrete-time dynamical systems, differential equations, and stochastic processes.
1.1 Biology and Dynamics.
1.2 Variables. Parameters. and Functions in Biology.
1.3 The Units and Dimensions of Measurements and Functions.
1.4 Linear Functions and Their Graphs.
1.5 Discrete-Time Dynamical Systems.
1.6 Analysis of Discrete-Time Dynamical Systems.
1.7 Expressing Solutions with Exponential Functions.
1.8 Oscillations and Trigonometry.
1.9 A Model of Gas Exchange in the Lung.
1.10 An Example of Nonlinear Dynamics.
1.11 An Excitable Systems I: The Heart.
2. Limits and Derivatives.
2.1 Introduction to Derivatives.
2.4 Computing Derivatives: Linear and Quadratic Functions.
2.5 Derivatives of Sums. Powers. and Polynomials.
2.6 Derivatives of Products and Quotients.
2.7 The Second Derivative. Curvature. and Acceleration.
2.8 Derivatives of Exponential and Logarithmic Functions.
2.9 The Chain Rule.
2.10 Derivatives of Trigonometric Functions.
3. Applications of Derivatives and Dynamical Systems.
3.1 Stability and the Derivative.
3.2 More Complicated Dynamics.
3.4 Reasoning About Functions.
3.5 Limits at Infinity.
3.6 Leading Behavior and L''Hopital''s Rule.
3.7 Approximating Functions with Lines and Polynomials.
3.8 Newton''s Method.
3.9 Panting and Deep Breathing.
4. Differential Equations. Integrals. and Their Applications.
4.1 Differential Equations.
4.2 Solving Pure-Time Differential Equations.
4.3 Integration of Special Functions. Integration by Substitution. by Parts. and by Partial Fractions.
4.4 Integrals and Sums.
4.5 Definite and Indefinite Integrals.
4.6 Applications of Integrals.
4.7 Improper Integrals.
5. Analysis of Autonomous Differential Equations.
5.1 Basic Differential Equations.
5.2 Equilibria and Display of Autonomous Differential Equations.
5.3 Stable and Unstable Equilibria.
5.4 Solving Autonomous Differential Equations.
5.5 Two Dimensional Differential Equations.
5.6 The Phase Plane.
5.7 Solutions in the Phase Plane.
5.8 The Dynamics of a Neuron.
6. Probability Theory and Descriptive Statistics.
6.1 Introduction to Probabilistic Models.
6.2 Stochastic Models of Diffusion and Genetics.
6.3 Probability Theory.
6.4 Conditional Probability.
6.5 Independence and Markov Chains.
6.6 Displaying Probabilities.
6.7 Random Variables.
6.8 Descriptive Statistics.
6.9 Descriptive Statistics for Spread.
7. Probability Models.
7.1 Joint Distributions.
7.2 Covariance and Correlation.
7.3 Sums and Products of Random Variables.
7.4 The Binomial Distribution.
7.5 Applications of the Binomial Distribution.
7.6 Waiting Times: Geometric and Exponential Distributions.
7.7 The Poisson Distribution.
7.8 The Normal Distribution.
7.9 Applying the Normal Approximation.
8. Introduction to Statistical Reasoning.
8.1 Statistics: Estimating Parameters.
8.2 Confidence Limits.
8.3 Estimating the Mean.
8.4 Hypothesis Testing.
8.5 Hypothesis Testing: Normal Theory.
8.6 Comparing Experiments: Normal Theory.
8.7 Analysis of Contingency Tables and Goodness of Fit.
8.8 Hypothesis Testing with the Method of Support.
Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.
This online instructor database offers complete worked solutions to all exercises in the text, allowing you to create customized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class. Vist Cengage.com/solutionbuilder for more information.
Instructor's Solutions Manual