introduction To Discrete Dynamical Systems | |

Biology And Dynamics Growth | |

Models of Malaria | |

Maintenance: Models of Neurons | |

Replication: Models of Genetics | |

Types of Dynamical Systems | |

updating Functions: Describing Growth | |

A Model Population: Bacterial Growth | |

A Model Organism: A Growing Tree | |

Functions: Terminology and Graphs | |

Exercises | |

units And Dimensions | |

Converting Between Units | |

Translating Between Dimensions | |

Checking: Dimensions and Estimation | |

Exercises | |

Linear Functions And Their Graphs | |

Proportional Relations | |

The Equation of a Line | |

Finding Equations and Graphing Lines | |

Inverse Functions: Looking Backward | |

Exercises | |

Finding Solutions: Describing The Dynamics | |

Bacterial Population Growth | |

Solving for Tree Height | |

Composition of Functions | |

Exercises | |

Solutions And Exponential Functions | |

Bacterial Population Growth in General | |

Laws of Exponents and Logs | |

Expressing Results with Exponentials | |

Exercises | |

Power Functions And Allometry | |

Power Relations and Exponential Growth | |

Power Relations and Lines | |

Power Relations in Biology: Shape and Flight | |

Exercises | |

Oscillations And Trigonometry | |

Sine and Cosine: A Review | |

Describing Oscillations with the Cosine | |

More Complicated Shapes | |

Exercises | |

Modeling And Cobwebbing | |

A Model of the Lungs | |

The Lung Updating Function | |

Cobwebbing: A Graphical Solution Technique | |

Exercises | |

Equilibria | |

Equilibria: Graphical Approach | |

Equilibria: Algebraic Approach | |

Equilibria: Algebra Involving Parameters | |

Exercises | |

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 | |

Limits And Derivatives | |

Differential Equations | |

Bacterial Growth Re-Measured | |

Rates of Change | |

The Limit | |

Exercises | |

Limits Limits of Functions | |

Applying the Mathematical Definition of a Limit | |

Properties of Limits | |

Exercises | |

More Limits | |

Left and Right-Hand Limits | |

Infinite Limits | |

Functions with More Complicated Limits | |

Exercises | |

Continuity | |

Continuous Functions | |

Properties of Continuous Functions | |

Input and Output Tolerances | |

Exercises | |

Computing Derivatives | |

The Derivative in General | |

Linear and Quadratic Derivatives | |

Derivatives and Graphs | |

Exercises | |

Derivatives Of Sums And Products | |

Derivatives of Sums | |

Derivatives of Products | |

Special Causes and Examples | |

Exercises | |

Derivatives Of Powers And Quotients | |

Derivatives of Power Functions | |

The Quotient Rule | |

The Power Rule: Negative Powers | |

Exercises | |

Derivatives Of Special Functions | |

The Derivative of the Exponential Function | |

The Derivative of the Natural Logarithm | |

The Derivatives of Trigonometric Functions | |

Exercises | |

The Chain Rule | |

The Derivative of a Composite Function | |

Derivatives of Inverse Functions | |

Application of the Chain Rule | |

Exercises | |

Applications Of Derivatives And Dynamical Systems | |

Approximating Functions | |

Approximating Functions; Examples | |

The Tangent Line in Deviation Form | |

Comparison with Other Linear Approximations | |

Exercises | |

Stability And The Derivative | |

Motivation | |

An Unusual Equilibrium | |

Computing Slopes at Equilibria | |

Exercises | |

Derivatives And Dynamics | |

Qualitative Dynamical Systems | |

The Multiplier | |

The Logistic Dynamical System | |

Exercises | |

Maximization | |

Types of Maxima | |

The Second Derivative | |

Maximizing Harvest | |

Exercises | |

Reasoning About Functions | |

Reasoning About Continuous Functions | |

Reasoning About Maximization | |

Rolle''s Theorem and the Mean Value Theorem | |

Exercises | |

Limits At Infinity | |

The Behavior of Functions at Infinity | |

Application to Absorption Functions | |

Limits of Sequences | |

Exercises | |

Leading Behavior and L''Hopital''s Rule Leading Behavior of Functions at Infinity | |

Leading Behavior of Functions at 0 | |

L''Hopital''s Rule | |

Exercises | |

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 | |

Panting And Deep Breathing | |

Breathing at Different Rates | |

Deep Breathing | |

Panting | |

Exercises | |

The Method Of Least Squares | |

Differential Equations, Integrals, And Their Applications | |

Differential Equations | |

Differential Equations: Examples and Terminology | |

Euler''s Method: Pure-Time | |

Euler''s Method: Autonomous | |

Exercises | |

Basic Differential Equations | |

Newton''s Law of Cooling | |

Diffusion Across a Membrane | |

A Continuous Time Model of Selection | |

Exercises | |

The Antiderivative | |

Pure-Time Differential Equations | |

Rules for Antiderivatives | |

Solving Polynomial Differential Equations | |

Exercises | |

Special Functions And Substitution | |

Integrals of Special Functions | |

The Chain Rule and Integration | |

Getting Rid of Excess Constants | |

Exercises | |

Integrals And Sums | |

Approximating Integrals with Sums | |

Approximating Integrals in General | |

The definite Integral | |

Exercises | |

Definite And Indefinite Integrals | |

The Fundamental Theorem of Calculus | |

The Summation Property of Definite Integrals | |

General Solution | |

Exercises | |

applications Of Integrals | |

Integrals and Areas | |

Integrals and Averages | |

Integrals and Mass | |

Exercises | |

Improper Integrals | |

Infinite Limits of Integration | |

Improper Integrals: Examples | |

Infinite Integrands | |

Exercises | |

Analysis Of Differential Equations | |

Autonomous Differential Equations | |

Review of Autonomous Differential Equations | |

Equilibria | |

Display of Differential Equations | |

Exercises | |

Stable And Unstable Equilibria | |

Recognizing Stable and Unstable Equilibria | |

Applications of the Stability Theorem | |

A Model of a Disease | |

Exercises | |

Solving Autonomous Equations | |

Separation of Variables | |

Pure-Time Equations Revisited | |

Applications of Separation of Variables | |

Exercises | |

Two Dimensional Equations | |

Predator-Prey Dynamics | |

Newton''s Law of Cooling | |

Euler''s Method | |

Exercises | |

The Phase-Plane | |

Equilibria and Nullclines: Predator-Prey Equations | |

Equilibria and Nullclines: Selection Equations | |

Equilibria and Nullclines: Newton''s Law of Cooling | |

Exercises | |

Solutions In The Phase-Plane | |

Euler''s Method in the Phase-Plane | |

Direction Arrows: Predator-Prey Equations | |

More Direction Arrows | |

Exercises | |

The Dynamics Of A Neuron | |

A Mathematician''s View of a Neuron | |

The Mathematics of Sodium Channels | |

The FitzHugh-Nagumo Equations | |

Exercises | |

Probability Theory And Descriptive Statistics | |

Probabilistic Models | |

Probability and Statistics | |

Stochastic Population Growth | |

Markov Chains | |

Exercises | |

Stochastic Models Of Diffusion | |

Stochastic Diffusion | |

Exercises | |

Stochastic Models Of Genetics | |

The Genetics of Inbreeding | |

The Dynamics of Height | |

Blending Inheritance | |

Exercises | |

Probability Theory | |

Sample Spaces and Events | |

Set Theory | |

Assigning Probabilities to Events | |

Exercises | |

Conditional Probability | |

The Law of Total Probability | |

Bayes'' Theorem and the Rare Disease Example | |

Exercises | |

Independence And Markov Chains | |

Independence | |

The Multiplication Rule for Independent Events | |

Markov Chains and Conditional Probability | |

Exercises | |

Displaying Probabilities | |

Probability and Cumulative Distributions | |

The Probability Density Function | |

The cumulative distribution function | |

Exercises | |

Random Variables | |

Types of Random Variable | |

Expectation: Discrete Case | |

Expectation: Continuous Case | |

Exercises | |

Descriptive Statistics | |

The Median | |

The Mode | |

The Geometric Mean | |

Exercises | |

Descriptive Statistics For Spread | |

Range And Percentiles | |

Mean Absolution Deviation | |

Variance | |

Exercises | |

Probability Models | |

Joint Distributions | |

Marginal Probability Distributions | |

Joint Distributions and Conditional Distributions | |

Exercises | |

Covariance And Correlation | |

Covariance | |

Correlation | |

Perfect Correlation | |

Exercises | |

Sums And Products Of Random Variables | |

Expectation of a Sum | |

Expectation of a Product | |

Variance of a Sum | |

Exercises | |

The Binomial Distribution The | |

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