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

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