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| Discrete-Time Dynamical Systems | |
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| Biology and Dynamics | |
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| Growth: Models of Malaria | |
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| Maintenance: Models of Neurons | |
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| Replication: Models of Genetics | |
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| Types of Dynamical Systems | |
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| Variables and Functions | |
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| Describing Measurements with Variables, Parameters and Graphs | |
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| Describing Relations Between Measurements with Functions | |
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| Combining Functions | |
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| Units and Dimensions | |
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| Converting Between Units | |
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| Translating Between Dimensions | |
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| Checking: Dimensions and Estimation | |
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| Linear Functions and their Graphs | |
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| Proportional Relations | |
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| Linear Functions and the Equation of a Line | |
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| Finding Equations and Graphing Lines | |
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| Solving Equations Involving Lines | |
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| Finding Inverse Functions | |
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| Discrete-Time Dynamical Systems | |
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| Discrete-time Dynamical Systems and Updating Functions | |
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| Manipulating Updating Functions | |
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| Discrete-Time Dynamical Systems: Units and Dimensions | |
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| Solutions | |
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| Analysis of Discrete-Tie Dynamical Systems | |
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| Cobwebbing: A Graphical Solution Technique | |
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| Equilibria: Graphical Approach | |
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| Equilibria: Algebraic Approach | |
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| Solutions and Exponential Functions | |
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| Bacterial Population Growth in General | |
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| Laws of Exponents and Logs | |
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| Expressing Results with Exponentials | |
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| Oscillations and Trigonometry | |
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| Sine and Cosine: A Review | |
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| Describing Oscillations with the Cosine | |
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| More Complicated Shapes | |
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| A Model of Gas Exchange in the Lung | |
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| A Model of the Lungs | |
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| The Lung System in General | |
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| Lung Dynamics with Absorption | |
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| An Example of Nonlinear Dynamics | |
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| A Model of Selection | |
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| The Discrete-Time Dynamical System and Equilibria | |
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| Stable and Unstable Equilibria | |
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| Excitable Systems I: The Heart | |
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| A Simple Heart | |
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| Second Degree Block | |
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| The Wenckeback Phenomenon | |
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| Limits and Derivatives | |
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| Introduction to Derivatives | |
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| The Average Rate of Change | |
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| Instantaneous Rates of Change | |
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| Limits and Derivatives | |
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| Differential Equations: A Preview | |
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| Limits | |
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| Limits of Functions | |
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| Left and Right-Hand Limits | |
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| Properties of Limits | |
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| Infinite Limits | |
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| Continuity | |
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| Continuous Functions | |
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| Input and Output Tolerances | |
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| Hysteresis | |
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| Computing Derivatives | |
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| Differentiable Functions | |
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| The Derivative of a Linear Function | |
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| A Quadratic Function | |
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| Derivatives of Sums, Powers, and Polynomials | |
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| The Sum Rule | |
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| Derivatives of Power Functions | |
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| Derivatives of Polynomials | |
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| Derivatives of Products and Quotients | |
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| The Product Rule | |
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| Special Cases and Examples | |
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| The Quotient Rule | |
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| The Second Derivative | |
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| The Second Derivative | |
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| Using the Second Derivative for Graphing | |
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| Acceleration | |
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| Exponentials snd Logarithms | |
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| The Exponential Function | |
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| The Natural Logarithm | |
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| Applications | |
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| The Chain Rule | |
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| The Derivative of a Composite Function | |
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| Derivatives of Inverse Functions | |
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| Applications | |
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| Derivatives of Trigonometric Functions | |
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| Deriving the Derivatives of Sine and Cosine | |
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| Other Trigonometric Functions | |
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| Applications | |
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| Derivatives and Dynamical Systems | |
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| Stability and the Derivative | |
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| Motivation | |
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| Stability and the Slope | |
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| Evaluating Stability with the Derivative | |
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| More Complex Dynamics | |
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| The Logistic Dynamical System | |
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| Qualitative Dynamical Systems | |
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| Analysis of the Logistic Dynamical System | |
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| Maximization | |
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| Minima and Maxima | |
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| Maximizing Food Intake Rate | |
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| Maximizing Fish Harvest | |
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| Reasoning About Functions | |
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| The Intermediate Value Theorem | |
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| Maximization: The Extreme Value Theorem | |
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| Rolle's Theorem and the Mean Value Theorem | |
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| Limits at Infinity | |
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| The Behavior of Functions at Infinity | |
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| Application to Absorption Functions | |
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| Limits of Sequences | |
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| Leading Behavior and L'hopital's Rule | |
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| Leading Behavior of Functions at Infinity | |
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| Leading Behavior of Functions at 0 | |
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| The Method of Matched Leading Behaviors | |
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| L'Hopital's Rule | |
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| Approximating Functions | |
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| The Tangent and Secant Lines | |
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| Quadratic Approximation | |
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| Taylor Polynomials | |
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| Newton's Method | |
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| Finding the Equilibrium | |
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| Newton's Method | |
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| Why Newton's Method Works and When it Fails | |
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| Panting and Deep Breathing | |
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| Breathing at Different Rates | |
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| Deep Breathing | |
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| Panting | |
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| Intermediate Optimum | |
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| Differential Equations, Integrals, and their Applications | |
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| Differential Equations | |
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| Differential Equations: Examples and Terminology | |
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| Graphical Solution of Pure-Time Differential Equations | |
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| Euler's Method: Pure-Time | |
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| Antiderivatives and Indefinite Integrals | |
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| Pure-Time Differential Equations | |
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| Rules for Antiderivatives | |
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| Solving Polynomial Differential Equations | |
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| Special Functions, Substitution, and Parts | |
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| Integrals of Special Functions | |
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| The Chain Rule and Integration | |
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| Using Substitution to Eliminate Constants | |
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| Integrals and Sums | |
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| Approximating Integrals with Sums | |
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| Approximating Integrals in General | |
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| The Definite Integral | |
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| Definite and Indefinite Integrals | |
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| The Fundamental Theorem of Calculus | |
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| The Summation Property of Definite Integral | |
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| Proof of the Fundamental Theorem of Calculus | |
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| Applications of Integrals | |
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| Integrals and Areas | |
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| Integrals and Averages | |
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| Integrals and Mass | |
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| Improper Integrals | |
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| Infinite Limits of Integration | |
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| Improper Integrals: Examples | |
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| Infinite Integrands | |
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| Autonomous Differential Equations | |
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| Autonomous Differential Equations | |
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| Review of Autonomous Differential Equations | |
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| Newton's Law of Cooling | |
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| Diffusion Across a Membrane | |
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| A Continuous Time Model of Competition | |
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| The Phase-Line Diagram | |
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| Equilibria | |
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| Display of Differential Equations | |
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| Stable and Unstable Equilibria | |
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| Recognizing Stable and Unstable Equilibria | |
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| Applications of the Stability Theorem | |
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| A Model of a Disease | |
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| Solving Autonomous Equations | |
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| Separation of Variables | |
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| Pure-Time Equations Revisited | |
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| Applications of Separation of Variables | |
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| Two Dimensional Equations | |
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| Predator-Prey Dynamics | |
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| Newton's Law of Cooling | |
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| Euler's Method | |
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| The Phase-Plane | |
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| Equilibria and Nullclines: Predator-Prey Equations | |
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| Equilibria and Nullclines: Competition Equations | |
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| Equilibria and Nullclines: Newton's Law of Cooling | |
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| Solutions in the Phase-Plane | |
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| Euler's Method in the Phase-Plane | |
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| Direction Arrows: Predator-Prey Equations | |
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| Direction Arrows for the Competition Equations | |
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| Directions Arrows for newton's Law of Cooling | |
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| The Dynamics of a Neuron | |
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| A Mathematician's View of a Neuron | |
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| The Mathematics of Sodium Channels | |
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| The Mathematics of Sodium Channel Blocking | |
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| The FitzHugh-Nagumo Equations | |
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| Weak Channel Blocking Mechanism | |
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| The Effects of Constant Applied Current | |
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| Probability Theory and Statistics | |
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| Introduction to Probabilistic Models | |
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| Probability and Statistics | |
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| Stochastic Population Growth: Stochastic Reproduction | |
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| Stochastic Population Growth: Stochastic Immigration | |
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| Markov Chains | |
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| Stochastic Models of Diffusion and Genetics | |
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| Stochastic Diffusion | |
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| The Genetics of Inbreeding | |
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| The Dynamics of Height | |
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| Blending Inheritance | |
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| Probability Theory | |
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| Sample Spaces and Events | |
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| Set Theory | |
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| Assigning Probabilities to Events | |
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| Exercises | |
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| Conditional Probability | |
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| Conditional Probability | |
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| The Law of Total Probability | |
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| Bayes' Theorem and the Rare Disease Example | |
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| Independence and Markov Chains | |
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| Independence | |
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| The Multiplication Rule for Independent Events | |
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| Markov Chains and Conditional Probability | |
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| Displaying Probabilities | |
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| Probability and Cumulative Distributions | |
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| The Probability Density Function: Derivation | |
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| Using the Probability Density Function | |
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| The Cumulative Distribution Function | |
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| Random Variables | |
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| Types of Random Variable | |
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| Expectation: Discrete Case | |
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| Expectation: Continuous Case | |
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| Descriptive Statistics | |
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| The Median | |
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| The Mode | |
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| The Geometric Mean | |
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| Descriptive Statistics for Spread | |
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| Range And Percentiles | |
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| Mean Absolute Deviation | |
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| Variance | |
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| The Coefficient of Variation | |
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| Supplementary Problems For Chapter 6 | |
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| Probability Models | |
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| Joint Distributions | |
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| Joint Distributions | |
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| Marginal Probability Distributions | |
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| Joint Distributions and Conditional Distributions | |
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| Covariance and Correlation | |
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| Covariance | |
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| Correlation | |
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| Perfect Correlation | |
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| Sums and Products of Random Variables | |
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| Expectation of a Sum | |
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| Expectation of a Product | |