Practical Analysis in One Variable

* Preface * Introduction * I. Numbers and Functions, Sequences and Limits * Mathematical Modeling * Natural Numbers Just Aren't Enough * Infinity and Mathematical Induction * Rational Numbers * Functions * Polynomials * Functions, Functions, and More Functions * Lipschitz Continuity * Sequences and Limits * Solving the Muddy Yard Model * Real Numbers * Functions of Real Numbers * The Bisection Algorithm * Inverse Functions * Fixed Points and Contraction Maps * II. Differential and Integral Calculus * The Linearization of a Function at a Point * Analyzing the Behavior of a Population Model * Interpretations of the Derivative * Differentiability on Intervals * Useful Properties of the Derivative * The Mean Value Theorem * Derivatives of Inverse Functions * Modeling with Differential Equations * Antidifferentiation * Integration * Properties of the Integral * Applications of the Integral * Rocket Propulsion and the Logarithm *Constant Relative Rate of Change and the Exponential * A Mass-Spring System and the Trigonometric Functions * Fixed Point Iteration and Newton's Method * Calculus Quagmires * III. You Want Analysis? We've Got Your Analysis Right Here * Notions of Continuity and Differentiability * Sequences of Functions * Relaxing Integration * Delicate Limits and Gross Behavior * The Weierstrass Approximation Theorem * The Taylor Polynomial * Polynomial Interpolation * Nonlinear Differential Equations * The Picard Iteration * The Forward Euler Method * A Conclusion or an Introduction? * References * Index *