Unlock the mysteries of Calculus with a fresh approach rooted in simplicity and historical insight. This book reintroduces a nearly forgotten idea from René Descartes (1596–1650), showing how the fundamental concepts of Calculus can be understood using just basic algebra. Starting with rational functions — the core of early Calculus — this method allows the reader to grasp the rules for derivatives without the intimidating concepts of limits or real numbers, making the subject more accessible than ever.
But the journey doesn't stop there. While attempting to apply this algebraic approach to exponential functions, the reader will encounter the limitations of simple methods, revealing the necessity for more advanced mathematical tools. This natural progression leads to the discovery of continuity, the approximation process, and ultimately, the introduction of real numbers and limits. These deeper concepts pave the way for understanding differentiable functions, seamlessly bridging the gap between elementary algebra and the profound ideas that underpin Calculus.
Whether you're a student, educator, or math enthusiast, this book offers a unique pathway to mastering Calculus. By connecting historical context with modern mathematical practice, it provides a richer, more motivating learning experience. For those looking to dive even deeper, the author's 2015 book, What is Calculus? From Simple Algebra to Deep Analysis, is the perfect next step.
Contents:
- Preface
- Why Do We Need a New Approach?
- The New Approach
- About the Author
- The Main Characters:
- The Rational Numbers
- Functions and Their Graphs
- Linear Functions and Slope
- Simple Algebra and Tangents:
- Quadratic Equations and Functions
- Double Roots and Tangents
- Motion with Variable Speed
- Tangents to Graphs of Polynomials
- Simple Differentiation Rules for Polynomials
- The Differential Calculus of Rational Functions:
- Rational Expressions and Functions
- Tangents and Simple Differentiation Rules
- Product and Quotient Rule
- Continuity and Approximation of Derivatives:
- Local Boundedness and Continuity
- Rates of Change
- Approximation of Algebraic Derivatives
- A Look Beyond Algebraic Functions
- Exercises
- The Heart of Real Analysis:
- Completeness of the Real Numbers
- Limits and Continuity
- Exponential Functions for Real Numbers
- Derivatives of Exponential Functions
- Differentiable Functions
- Some Basic Properties of Differentiable Functions
- Applications of Derivatives: A Brief Introduction:
- Acceleration and Motion with Constant Acceleration
- The Inverse Problem and Antiderivatives
- Exponential Models
- 'Explosive Growth' Models
- Periodic Motions
- Epilogue
- Index
Readership: High school math teachers and high school students interested in the introduction to calculus; mathematics educators in volved in math curriculum development; first year college students, college calculus instructors; college math students; mathematics historians and general science readers.
