Calculus: The Fundamental Concepts, Through Their History

Course Description

Summary

This class focuses on what is most intellectually interesting about calculus: the problems it was invented to solve, the fundamental ideas, and the interconnection between the ideas. The class approaches integration, infinite series, differentials, and differential equations, in a unified way. It focuses on concepts and interconnections. In the process, the class builds skills in what is often most important for users of calculus: how to recognize when a problem is a calculus problem, and how to set it up; and, going in the other direction, how to interpret a given calculus solution of a problem (such as a differential equation model).

The approach of the class is historical: to look at the original problems the calculus was attempting to solve, to follow the invention of the different ideas, and to follow the critical controversies around their meaning. In this way, the motivation and the meaning is emphasized. However, this is not a history class; rather, the focus is on the mathematics, motivated by, and embedded in, the historical narrative.

In order to cover this much material in one term, computational techniques are de-emphasized. Further techniques and applications, which would normally be covered in a first calculus sequence, will appear in following mathematics courses.

This is an introductory course; no previous experience with calculus is assumed. However, if you have taken some calculus already, and want a better understanding of what is actually going on, this class will have enough new material to be valuable for you, even if you have taken some fairly advanced calculus. The only prerequisite for the class is a good comfort level with high-school algebra. Some exposure to trigonometric functions, exponential functions, and logs would be helpful, but they will all be re-defined and re-derived in the class. In particular, a "pre-calculus" course, while possibly helpful, is not a prerequisite for this class.