This book attempts to place the basic ideas of real analysis and numerical analysis together in an applied setting that is both accessible and motivational to young students. The essentials of real analysis are presented in the context of a fundamental problem of applied mathematics, which is to approximate the solution of a physical model. The framework of existence, uniqueness, and methods to approximate solutions of model equations is sufficiently broad to introduce and motivate all the basic ideas of real analysis. The book includes background and review material, numerous examples, visualizations and alternate explanations of some key ideas, and a variety of exercises ranging from simple computations to analysis and estimates to computations on a computer. The book can be used in an honor calculus sequence typically taken by freshmen planning to major in engineering, mathematics, and science, or in an introductory course in rigorous real analysis offered to mathematics majors. Donald Estep is Professor of Mathematics at Colorado State University. He is the author of Computational Differential Equations, with K. Eriksson, P. Hansbo and C. Johnson (Cambridge University Press 1996) and Estimating the Error of Numerical Solutions of Systems of Nonlinear Reaction-Diffusion Equations with M. Larson and R. Williams (A.M.S. Memoirs, 2000), and recently co-edited Collected Lectures on the Preservation of Stability under Discretization, with Simon Tavener (S.I.A.M., 2002), as well as numerous research articles. His research interests include computational error estimation and adaptive finite element methods, numerical solution of evolutionary problems, and computational investigation of physical models
CONTENT
Numbers and Functions, Sequences and Limits -- Mathematical Modeling -- Natural Numbers Just Arenโ{128}{153}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 -- 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 Newtonโ{128}{153}s Method -- Calculus Quagmires -- You Want Analysis? Weโ{128}{153}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
