Calculus: A New Approach For Schools That Starts With Simple Algebra

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.

R Michael Range earned his Diplom in Mathematik at the University of Göttingen, Germany, in 1968. A Fulbright Fellowship brought him to the United States, where he earned a PhD at UCLA in 1971. He has held academic positions at Yale University, University of Washington, and the State University of New York at Albany, as well as research positions at institutes in Bonn, Stockholm, Barcelona, and Berkeley. He retired in 2015. Range is an expert in multidimensional complex analysis. He has published numerous research articles in that area, and he wrote Holomorphic Functions and Integral Representations in Several Complex Variables (Springer 1986 and 1998), a widely used introduction to the field. He has lectured in many countries, including mini courses at the Chinese Academy of Sciences in Beijing, POSTECH in South Korea, and in Trondheim, Norway. In 2021, he presented a popular lecture via zoom at an international conference in Mexico City. In recent years, he has written historical and expository articles on multidimensional complex analysis. He also has devoted much time to developing a new, elementary introduction to Calculus, culminating in the book What is Calculus? From Simple Algebra to Deep Analysis (World Scientific Publishing, 2015). For more details, see www.albany.edu/~range/.