Early Transcendental Analysis

TL;DR

This paper provides a rigorous foundation for the early transcendental approach to exponential and logarithmic functions in calculus, using mainly the supremum property of real numbers to justify the methods.

Contribution

It offers a rigorous treatment of early transcendental calculus methods suitable for introductory analysis courses, avoiding advanced tools used in prior justifications.

Findings

Rigorous justification of exponential and logarithmic functions

Use of supremum property for foundational analysis

Clarification of early transcendental methods

Abstract

In Early Transcendentals (The American Mathematical Monthly, Vol. 104, No 7) Steven Weintraub presents a rigorous justifcation of the "early transcendental" calculus textbook approach to the exponential and logarithmic functions. However, he uses tools such as term-by-term differentiation of infinite series. We present a rigorous treatment of the early transcendental approach suitable for a first course in analysis, using mainly the supremum property of the real numbers.