---
abstract: 'This article exemplifies a novel approach to the teaching of introductory differential calculus using the modern notion of “infinitesimal” as opposed to the traditional approach using the notion of “limit”. I illustrate the power of the new approach with a discussion of the derivatives of the sine and cosine functions.'
author:
- |
    JACK L. URETSKY\
    High Energy Physics Division, Argonne National Laboratory\
    (jlu@hep.anl.gov)
title: 'INTRODUCTORY CALCULUS FROM THE VIEWPOINT OF NON-STANDARD ANALYSIS - DERIVATIVE OF SINE AND COSINE'
---

-.68in - 1.0in 6.5in 8.in

OVERVIEW
========

Some teachers of mathematics may be surprised to learn that the concept of limits was a rather late (d’Alembert, 1754) addition to the underpinnings of the calculus. The inventors of calculus worked directly with concepts such as “infinitely little lines” (Newton, in Cohen and Westfall’s [*Newton*]{}, Norton, New York 1995, p.381), presumably adopting the most intuitively congenial approach to the subject. This fact suggests that it is counter-intuitive to introduce students to the derivative as the outcome of a limiting process. In the words of one mathematics teacher:

> As someone who has spent a good portion of his adult life teaching calculus courses for a living, I can tell you 