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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, 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.
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 / Second-Degree Block / The Wenckebach Phenomenon / Exercises PART II: LIMITS AND DERIVATIVES 12. DIFFERENTIAL EQUATIONS Bacterial Growth Re-Measured / 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 Right-Hand 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: Pure-Time / 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 Pure-Time 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 / Pure-Time Equations Revisited / Applications of Separation of Variables / Exercises 42. TWO DIMENSIONAL EQUATIONS Predator-Prey Dynamics / Newton's Law of Cooling / Euler's Method / Exercises 43. THE PHASE-PLANE Equilibria and Nullclines: Predator-Prey Equations / Equilibria and Nullclines: Selection Equations / Equilibria and Nullclines: Newton's Law of Cooling / Exercises 44. SOLUTIONS IN THE PHASE-PLANE Euler's Method in the Phase-Plane / Direction Arrows: Predator-Prey Equations / More Direction Arrows / Exercises 45. THE DYNAMICS OF A NEURON A Mathematician's View of a Neuron / The Mathematics of Sodium Channels / The FitzHugh-Nagumo 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 P-Values 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