Calculus: A new approach for schools that starts with simple algebra

Abstract

We outline a novel approach to tangents and derivatives that does not use any limits. Instead, it uses elementary algebraic concepts related to the quadratic equation, and therefore fits right into the school curriculum. Adding an elementary estimate to the central algebraic factorization naturally leads to the concept of continuity and thereby reveals that the algebraic derivative can also be captured by an approximation process. This turns out to be critical for handling non-algebraic functions, such as exponential functions. In order to capture the new elusive limits, students recognize that one needs to expand the familiar rational numbers to the much more intriguing real numbers. The solution of the tangent problem for exponential functions leads to the general notion of a differentiable function, in a formulation that is the natural generalization of the algebraic version, and which has been known for over 70 years. This approach gradually proceeds from most elementary concepts to the heart of analysis, making it clear to students along the way why more sophisticated tools are needed, and providing motivation for the more advanced concepts that are indispensable for a proper understanding of calculus.