Differential Calculus: From Practice to Theory
Credit: Cover of "Differential Calculus: From Practice to Theory" textbook, adapted from a work by Crockett Johnson, used with permission
Resource Description
Differential Calculus: From Practice to Theory covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the foundational ideas (limits, continuity) are developed to replace infinitesimals, first intuitively then rigorously. This approach is more historically accurate than the usual development of calculus and, more importantly, it is pedagogically sound.
Learn moreHow We Got from There to Here: A Story of Real Analysis
Credit: Course image adapted from a figure by Eugene Boman and Robert Rogers and is licensed under CC BY-NC-SA 4.0
Resource Description
The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.
Learn moreTechniques of Calculus, I
Credit: Image adapted from a figure in Applied Calculus: Principles and Applications by Robert Gibbes Thomas, public domain
Resource Description
A Jupyter notebook companion for a first course in Calculus, including a review of Precalculus concepts.
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