What is Calculus‪?‬

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: Introduction Tangents to Circles Tangents to Parabolas Motion with Variable Speed Tangents to Graphs of Polynomials Rules for Differentiation More General Algebraic Functions Beyond Algebraic Functions The Cast: Functions of a Real Variable: Real Numbers Functions Simple Periodic Functions Exponential Functions Natural Operations on Functions Algebraic Operations and Functions Derivatives: How to Measure Change: Algebraic Derivatives by Approximation Derivatives of Exponential Functions Differentiability and Local Linear Approximation Properties of Continuous Functions Derivatives of Trigonometric Functions Simple Differentiation Rules Product and Quotient Rules Some Applications of Derivatives: Exponential Models The Inverse Problem and Antiderivatives "Explosive Growth" Models Acceleration and Motion with Constant Acceleration Periodic Motions Geometric Properties of Graphs An Algorithm for Solving Equations Applications to Optimization Higher Order Approximations and Taylor Polynomials The Definite Integral: The Inverse Problem: Construction of Antiderivatives The Area Problem More Applications of Definite Integrals Properties of Definite Integrals The Fundamental Theorem of Calculus Existence of Definite Integrals Reversing the Chain Rule: Substitution Reversing the Product Rule: Integration by Parts Higher Order Approximations, Part 2: Taylor's Theorem Excursion into Complex Numbers and the Euler Identity Readership: 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 beginning Accessible to readers with minimal prerequisites Fundamental concepts of analysis are only introduced when necessary, for example, when studying exponential functions Systematic 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 rules Stimulates thought and discussion about potential changes in the standard calculus curriculum