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More About This Textbook
Overview
Designed to help life sciences students understand the role mathematics has played in breakthroughs in epidemiology, genetics, statistics, physiology, and other biological areas, this text provides students with a thorough grounding in mathematics, the language, and 'the technology of thought' with which these developments are created and controlled.
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Table of Contents
PART I: INTRODUCTION TO DISCRETE DYNAMICAL SYSTEMS 1. BIOLOGY AND DYNAMICS Growth: Models of Malaria / Maintenance: Models of Neurons / Replication: Models of Genetics / Types of Dynamical Systems 2. UPDATING FUNCTIONS: DESCRIBING GROWTH A Model Population: Bacterial Growth / A Model Organism: A Growing Tree / Functions: Terminology and Graphs / Exercises 3. UNITS AND DIMENSIONS Converting Between Units / Translating Between Dimensions / Checking: Dimensions and Estimation / Exercises 4. LINEAR FUNCTIONS AND THEIR GRAPHS Proportional Relations / The Equation of a Line / Finding Equations and Graphing Lines / Inverse Functions: Looking Backward / Exercises 5. FINDING SOLUTIONS: DESCRIBING THE DYNAMICS Bacterial Population Growth / Solving for Tree Height / Composition of Functions / Exercises 6. SOLUTIONS AND EXPONENTIAL FUNCTIONS Bacterial Population Growth in General / Laws of Exponents and Logs / Expressing Results with Exponentials / Exercises 7. POWER FUNCTIONS AND ALLOMETRY Power Relations and Exponential Growth / Power Relations and Lines / Power Relations in Biology: Shape and Flight / Exercises 8. OSCILLATIONS AND TRIGONOMETRY Sine and Cosine: A Review / Describing Oscillations with the Cosine / More Complicated Shapes / Exercises 9. MODELING AND COBWEBBING A Model of the Lungs / The Lung Updating Function / Cobwebbing: A Graphical Solution Technique / Exercises 10. EQUILIBRIA Equilibria: Graphical Approach / Equilibria: Algebraic Approach / Equilibria: Algebra Involving Parameters / Exercises 11. NONLINEAR DYNAMICS A Model of Selection / The General Case and Equilibria / Stable and Unstable Equilibria / Exercises / A Simple Heart / SecondDegree Block / The Wenckebach Phenomenon / Exercises PART II: LIMITS AND DERIVATIVES 12. DIFFERENTIAL EQUATIONS Bacterial Growth ReMeasured / Rates of Change / The Limit / Exercises 13. LIMITS Limits of Functions / Applying the Mathematical Definition of a Limit / Properties of Limits / Exercises 14. MORE LIMITS Left and RightHand Limits / Infinite Limits / Functions with More Complicated Limits / Exercises 15. CONTINUITY Continuous Functions / Properties of Continuous Functions / Input and Output Tolerances / Exercises 16. COMPUTING DERIVATIVES The Derivative in General / Linear and Quadratic Derivatives / Derivatives and Graphs / Exercises 17. DERIVATIVES OF SUMS AND PRODUCTS Derivatives of Sums / Derivatives of Products / Special Causes and Examples / Exercises 18. DERIVATIVES OF POWERS AND QUOTIENTS Derivatives of Power Functions / The Quotient Rule / The Power Rule: Negative Powers / Exercises 19. DERIVATIVES OF SPECIAL FUNCTIONS The Derivative of the Exponential Function / The Derivative of the Natural Logarithm / The Derivatives of Trigonometric Functions / Exercises 20. THE CHAIN RULE The Derivative of a Composite Function / Derivatives of Inverse Functions / Application of the Chain Rule / Exercises PART III: APPLICATIONS OF DERIVATIVES AND DYNAMICAL SYSTEMS 21. APPROXIMATING FUNCTIONS Approximating Functions; Examples / The Tangent Line in Deviation Form / Comparison with Other Linear Approximations / Exercises 22. STABILITY AND THE DERIVATIVE Motivation / An Unusual Equilibrium / Computing Slopes at Equilibria / Exercises 23. DERIVATIVES AND DYNAMICS Qualitative Dynamical Systems / The Multiplier / The Logistic Dynamical System / Exercises 24. MAXIMIZATION Types of Maxima / The Second Derivative / Maximizing Harvest / Exercises 25. REASONING ABOUT FUNCTIONS Reasoning About Continuous Functions / Reasoning About Maximization / Rolle's Theorem and the Mean Value Theorem / Exercises 26. LIMITS AT INFINITY The Behavior of Functions at Infinity / Application to Absorption Functions / Limits of Sequences / Exercises 27. Leading Behavior and L''Hopital's Rule Leading Behavior of Functions at Infinity / Leading Behavior of Functions at 0 / L''Hopital's Rule / Exercises 28. NEWTON's METHOD Finding the Equilibrium of the Lung Model with Absorption / Newton's Method / Why Newton's Method Works and When it fails / Exercises 29. PANTING AND DEEP BREATHING Breathing at Different Rates / Deep Breathing / Panting / Exercises 30. THE METHOD OF LEAST SQUARES PART IV: DIFFERENTIAL EQUATIONS, INTEGRALS, AND THEIR APPLICATIONS 31. DIFFERENTIAL EQUATIONS Differential Equations: Examples and Terminology / Euler's Method: PureTime / Euler's Method: Autonomous / Exercises 32. BASIC DIFFERENTIAL EQUATIONS Newton's Law of Cooling / Diffusion Across a Membrane / A Continuous Time Model of Selection / Exercises 33. THE ANTIDERIVATIVE PureTime Differential Equations / Rules for Antiderivatives / Solving Polynomial Differential Equations / Exercises 34. SPECIAL FUNCTIONS AND SUBSTITUTION Integrals of Special Functions / The Chain Rule and Integration / Getting Rid of Excess Constants / Exercises 35. INTEGRALS AND SUMS Approximating Integrals with Sums / Approximating Integrals in General / The definite Integral / Exercises 36. DEFINITE AND INDEFINITE INTEGRALS The Fundamental Theorem of Calculus / The Summation Property of Definite Integrals / General Solution / Exercises 37. APPLICATIONS OF INTEGRALS Integrals and Areas / Integrals and Averages / Integrals and Mass / Exercises 38. IMPROPER INTEGRALS Infinite Limits of Integration / Improper Integrals: Examples / Infinite Integrands / Exercises PART V: ANALYSIS OF DIFFERENTIAL EQUATIONS 39. AUTONOMOUS DIFFERENTIAL EQUATIONS Review of Autonomous Differential Equations / Equilibria / Display of Differential Equations / Exercises 40. STABLE AND UNSTABLE EQUILIBRIA Recognizing Stable and Unstable Equilibria / Applications of the Stability Theorem / A Model of a Disease / Exercises 41. SOLVING AUTONOMOUS EQUATIONS Separation of Variables / PureTime Equations Revisited / Applications of Separation of Variables / Exercises 42. TWO DIMENSIONAL EQUATIONS PredatorPrey Dynamics / Newton's Law of Cooling / Euler's Method / Exercises 43. THE PHASEPLANE Equilibria and Nullclines: PredatorPrey Equations / Equilibria and Nullclines: Selection Equations / Equilibria and Nullclines: Newton's Law of Cooling / Exercises 44. SOLUTIONS IN THE PHASEPLANE Euler's Method in the PhasePlane / Direction Arrows: PredatorPrey Equations / More Direction Arrows / Exercises 45. THE DYNAMICS OF A NEURON A Mathematician's View of a Neuron / The Mathematics of Sodium Channels / The FitzHughNagumo Equations / Exercises PART VI: PROBABILITY THEORY AND DESCRIPTIVE STATISTICS 46. PROBABILISTIC MODELS Probability and Statistics / Stochastic Population Growth / Markov Chains / Exercises 47. STOCHASTIC MODELS OF DIFFUSION Stochastic Diffusion / Exercises 48. STOCHASTIC MODELS OF GENETICS The Genetics of Inbreeding / The Dynamics of Height / Blending Inheritance / Exercises 49. PROBABILITY THEORY Sample Spaces and Events / Set Theory / Assigning Probabilities to Events / Exercises 50. CONDITIONAL PROBABILITY The Law of Total Probability / Bayes'' Theorem and the Rare Disease Example / Exercises 51. INDEPENDENCE AND MARKOV CHAINS Independence / The Multiplication Rule for Independent Events / Markov Chains and Conditional Probability / Exercises 52. DISPLAYING PROBABILITIES Probability and Cumulative Distributions / The Probability Density Function / The cumulative distribution function / Exercises 53. RANDOM VARIABLES Types of Random Variable / Expectation: Discrete Case / Expectation: Continuous Case / Exercises 54. DESCRIPTIVE STATISTICS The Median / The Mode / The Geometric Mean / Exercises 55. DESCRIPTIVE STATISTICS FOR SPREAD Range And Percentiles / Mean Absolution Deviation / Variance / Exercises PART VII: PROBABILITY MODELS 56. JOINT DISTRIBUTIONS Marginal Probability Distributions / Joint Distributions and Conditional Distributions / Exercises 57. COVARIANCE AND CORRELATION Covariance / Correlation / Perfect Correlation / Exercises 58. SUMS AND PRODUCTS OF RANDOM VARIABLES Expectation of a Sum / Expectation of a Product / Variance of a Sum / Exercises 59. THE BINOMIAL DISTRIBUTION The Binomial Distribution Defined / Computing the Binomial / Binomial Distribution: The General Case / Exercises 60. APPLICATIONS OF THE BINOMIAL DISTRIBUTION Application to Genetics and Calculation of Mode / Application to Markov Chains: Definition of Cumulative Distribution / Applications to Diffusion / Exercises 61. EXPONENTIAL DISTRIBUTIONS The Geometric Distribution / The Exponential Distribution / The Memoryless Property / Exercises 62. THE POISSON DISTRIBUTION The Poisson Process / The Poisson Distribution in Space / The Poisson and the Binomial / Exercises 63. THE NORMAL DISTRIBUTION The Normal Distribution: An Example / The Central Limit Theorem for Sums / The Central Limit Theorem for Averages / Exercises 64. APPLYING THE NORMAL APPROXIMATION The Standard Normal Distribution / Normal Approximation of the Binomial / Normal Approximation of the Poisson / Exercises PART VIII: INTRODUCTION TO STATISTICAL REASONING 65. STATISTICS: ESTIMATING PARAMETERS Estimating the Binomial Proportion / Maximum Likelihood / Estimating a Rate / Exercises 66. CONFIDENCE LIMITS Exact Confidence Limits / Monte Carlo Method / Likelihood, Support, and Confidence Limits / Exercises 67. ESTIMATING THE MEAN Confidence Limits / Sample Variance and Standard Error / Exercises 68. HYPOTHESIS TESTING Hypothesis Testing: an Example / Power and Confidence Limits / Likelihood and the Method of Support / Exercises 69. HYPOTHESIS TESTING: NORMAL THEORY Computing PValues with the Normal Approximation / The power of Normal Tests / Likelihood and the Normal Distribution / Exercises 70. COMPARING EXPERIMENTS Unpaired Normal Distributions / Comparing Population Proportions / Likelihood / Exercises 71. REGRESSION Linear Regression / Using Linear Regression / The Theory of Linear Regression / Exercises