Calculus without Limit Theory

TL;DR

This paper introduces a new approach to calculus based on physical facts rather than limits, providing elementary definitions and proofs that align with traditional calculus for smooth functions, aiming to improve teaching methods.

Contribution

It offers a limit-free foundation for calculus using physical principles, simplifying the understanding and teaching of derivatives and integrals.

Findings

Defines derivatives and integrals without limits

Proves the fundamental theorem of calculus without auxiliary conditions

Aligns with traditional calculus for smooth functions

Abstract

This paper establishes calculus upon two physical facts: (1) any average velocity is always between two instantaneous velocities, and (2) the motion of an object is determined once its velocity has been determined. It directly defines derivative and definite integral on an ordered field, proves the fundamental theorem of calculus with no auxiliary conditions, easily reveals the common properties of derivatives, and obtains differentiation formulas for elementary functions. Further discussion shows that the new definitions are in accord with the traditional concepts for continuously differentiable functions. This is a result of the authors' research in the field of educational mathematics, which hopes to provide a more elementary and effective way to teach calculus.