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book cover

The Six Pillars of Calculus: Biology Edition

By Lorenzo Sadun. AMSTEXT/60.

book cover

The Six Pillars of Calculus: Business Edition

By Lorenzo Sadun. AMSTEXT/56.

Every fall hundreds of thousands of students take their calculus class, and thousands of instructors start teaching it. These instructors search through hundreds of textbooks trying to find one that is best for their students. So, why has the AMS decided to offer more books to choose from?

One answer might be that the needs of students who take calculus are constantly changing. In the second half of the 20th century calculus, as it was taught in colleges, became a purely mathematical discipline studying, mainly, functions and relations between them. Of course, applications were mentioned, but meaningful analysis of these applications was often insufficient. On the other hand, the majority of students taking calculus aspire to become not just mathematicians or engineers, but also biologists, medical doctors, economists, political scientists, or pursue other careers, and they do not see the benefits of learning this seemingly boring and irrelevant subject. The 1990s brought about new approaches to teaching calculus, including the book Calculus in Context: The Five College Calculus Project. The vision of the authors of this book was that calculus is “a language and a tool for exploring the whole fabric of science.”

The Six Pillars of Calculus: Biology Edition and The Six Pillars of Calculus: Business Edition by Lorenzo Sadun, published in 2023, may be viewed as further development of this approach. While presenting the same main ideas, topics, and techniques as many other calculus textbooks, motivation and illustrative examples come from biology and from business respectively. For example, Chapter 2 in both books describes the SIR model, first introduced in epidemiology. In the biology book it is viewed as a model of epidemics spread, and in the business version it is presented as a model of market penetration, describing how a new product wins the market.

Another important feature of the books is that the author presents all the material using the following six main principles (pillars): 1. Close is good enough. 2. Track the changes. 3. What goes up should stop before it comes down. 4. The whole is the sum of the parts. 5. One step at a time. 6. One variable at a time.

Each book starts with introductory Chapter 1 explaining what the six pillars of calculus are, and illustrating them with simple examples. The next two chapters explain how to build a good model of the phenomenon you want to analyze and what you can hope to achieve by analyzing it. In Chapters 4 and 5 the notion of the derivative is introduced and explained, whereas in Chapter 6 the author studies the construction and analysis of simple models using simple differential equations, again with examples drawn from business or biology respectively. In Chapters 7 and 8 the idea of the integral, the notion of anti-derivative, and the fundamental theorem of calculus are introduced. In Chapter 9 several methods of integration are presented including the idea of numerical integration. Chapter 10 brings up selected topics from multivariable calculus. Finally, in Chapter 11 Taylor series are introduced and the convergence radius of a power series is explained. Each chapter ends with a one-page chapter summary.

A very important component of both books is the exercises, about sixty exercises per chapter. Whereas some of them are relatively standard and aim at understanding the main techniques of calculus, many are based on examples from biology or business respectively.