What is Calculus? (ePub)

This unique book provides a new and well-motivated introduction to calculus and analysis, historically significant fundamental areas of mathematics that are widely used in many disciplines. It begins with familiar elementary high school geometry and algebra, and develops important concepts such as tangents and derivatives without using any advanced tools based on limits and infinite processes that dominate the traditional introductions to the subject. This simple algebraic method is a modern version of an idea that goes back to René Descartes and that has been largely forgotten. Moving beyond algebra, the need for new analytic concepts based on completeness, continuity, and limits becomes clearly visible to the reader while investigating exponential functions.The author carefully develops the necessary foundations while minimizing the use of technical language. He expertly guides the reader to deep fundamental analysis results, including completeness, key differential equations, definite integrals, Taylor series for standard functions, and the Euler identity. This pioneering book takes the sophisticated reader from simple familiar algebra to the heart of analysis. Furthermore, it should be of interest as a source of new ideas and as supplementary reading for high school teachers, and for students and instructors of calculus and analysis.Contents:Prelude to Calculus:IntroductionTangents to CirclesTangents to ParabolasMotion with Variable SpeedTangents to Graphs of PolynomialsRules for DifferentiationMore General Algebraic FunctionsBeyond Algebraic FunctionsThe Cast: Functions of a Real Variable:Real NumbersFunctionsSimple Periodic FunctionsExponential FunctionsNatural Operations on FunctionsAlgebraic Operations and FunctionsDerivatives: How to Measure Change:Algebraic Derivatives by ApproximationDerivatives of Exponential FunctionsDifferentiability and Local Linear ApproximationProperties of Continuous FunctionsDerivatives of Trigonometric FunctionsSimple Differentiation RulesProduct and Quotient RulesSome Applications of Derivatives:Exponential ModelsThe Inverse Problem and Antiderivatives"Explosive Growth" ModelsAcceleration and Motion with Constant AccelerationPeriodic MotionsGeometric Properties of GraphsAn Algorithm for Solving EquationsApplications to OptimizationHigher Order Approximations and Taylor PolynomialsThe Definite Integral:The Inverse Problem: Construction of AntiderivativesThe Area ProblemMore Applications of Definite IntegralsProperties of Definite IntegralsThe Fundamental Theorem of CalculusExistence of Definite IntegralsReversing the Chain Rule: SubstitutionReversing the Product Rule: Integration by PartsHigher Order Approximations, Part 2: Taylor's TheoremExcursion into Complex Numbers and the Euler IdentityReadership: Undergraduates, high school students, instructors and teachers, and scientifically literate readers with special interest in calculus and analysis.Key Features:This is the first, and so far, only book that implements a new and more elementary approach to calculus and analysis which differs from all standard treatments that place deep new concepts based on limits right at the beginningAccessible to readers with minimal prerequisitesFundamental concepts of analysis are only introduced when necessary, for example, when studying exponential functionsSystematic usage of a reformulation of the definition of differentiability that allows to simplify standard proofs of basic results, and in particular, makes it clear that the chain rule is really much simpler and more natural than the product and quotient rulesStimulates thought and discussion about potential changes in the standard calculus curriculum