Practical Analysis in One Variable

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. It 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. Students who work through the proofs and solve the practical problems in this book will develop a "hands-on" understanding of analysis that will serve them well in the future.

Includes supplementary material: sn.pub/extras

Contenu
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.- 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.- 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.