Presents a limit-free algebraic-geometric construction of derivatives for rational, exponential, logarithmic, trigonometric and inverse trigonometric functions based on tangent lines and local linear structure.
Calculus without Limit Theory
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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.
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2026 1verdicts
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A Limit-Free Algebraic-Geometric Construction of Derivatives for Elementary Functions
Presents a limit-free algebraic-geometric construction of derivatives for rational, exponential, logarithmic, trigonometric and inverse trigonometric functions based on tangent lines and local linear structure.