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Full Description
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
Tangents and Double Points; Derivatives by Algebra; Exponential Functions; Completeness of Real Numbers; The Base of the Natural Exponential and Logarithm Functions; Continuity of Functions; Differentiability; Chain Rule and Other Rules for Derivatives; Derivatives of Trigonometric Functions; Mean Value Inequality and Theorem; Basic Differential Equations; Motion with Constant Acceleration; Linear and Higher Order Approximations; The Antiderivative Problem; Definite Integrals; Fundamental Theorem of Calculus; Integrability of Monotonic Functions; Integrability of Functions with Bounded Derivative; Substitution; Integration by Parts; Taylor's Theorem; Analytic Functions; The Euler Identity;
