Calculus Set Free

In Calculus Set Free, C. Bryan Dawson offers a comprehensive introduction to the topics present in a typical Calculus 1 and Calculus 2 course, with one major twist: the theory of limits and of calculus that Dawson develops is built on infinitesimals and the hyperreal number system. Although this choice is a major divergence from typical calculus books, the text does not feel radical; it covers the usual calculus topics in the usual order, and stands on its own as a well-organized, well written text with many appropriate examples and exercises. The result is a high-quality Calculus textbook that offers an alternative approach to some of the theoretical underpinnings of the topic. 

 

The text’s subtitle is Infinitesimals to the Rescue, and hints at the importance that the hyperreal number system plays in this text. Eschewing the typical introduction to limits, Dawson starts the text by building a sufficiently robust theory of hyperreal numbers, and uses these hyperreals -- not epsilons and deltas -- to rigorously define limits. The key theoretical ingredients are introduced immediately, including an introduction to hyperreals (Chapter 1.1), the use of approximation in evaluating hyperreal numbers (Chapter 1.2), and the way in which hyperreal numbers can be plugged into functions with real-valued outputs (Chapter 1.3). Armed with these tools, the book moves forward as a relatively typical Calculus text, covering all of the usual topics in the usual order. The infinitesimals continue to appear, but their primary role is in the proofs present throughout the text, as well as in some of the early examples of derivative and integral computations. As in a typical text, the book eventually moves from early examples to shortcuts and applications (similar to how a typical calculus text navigates the limit definition of a derivative before switching to derivative shortcuts.) As a result, much of the text does not feel radically different from a regular Calculus textbook.

 

Because the infinitesimal approach is the major difference between this text and many others, one might expect the author to lay out a strong and robust argument for why this approach should be followed. In the “note to the instructor,” Dawson justifies this approach by claiming that (1) hyperreal numbers allow for a more intuitive approach to certain topics in calculus (such as comparing growth rates), (2) hyperreal numbers offer an approach that “corresponds more closely to the way our colleagues in other disciplines teach students to analyze their ideas,” and (3) the use of hyperreals allow for limits to be discussed in an algebraically simpler manner. I believe each of these justifications is certainly true to an extent, but I would love to see more said in defense of this new paradigm. My fear is that students who will struggle with the concept of limits (i.e., most students) will also struggle with the concept of the hyperreals and that the new approach might obscure the ``limiting behavior’’ of derivatives and Riemann sums, even as it simplifies the algebra used. I would love the author to speak more to this concern in their justification.

 

In summary: Calculus Set Free is a well-written and self-contained text which offers a novel and mathematically rigorous approach to the topics typically present in Calculus 1 and 2. The text is largely successful in what it sets out to accomplish, and teachers interested in offering an introduction to Calculus built on an alternative theoretical approach should consider this text.

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John Ross is an assistant professor of mathematics at Southwestern University.