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Single Variable Calculus: A First Step

Yunzhi Zou
Walter de Gruyter
Publication Date: 
Number of Pages: 
De Gruyter Textbook
[Reviewed by
Tom Schulte
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This is an efficiently compressed entry-level calculus introduction. The core concepts and applications of limits, continuity, derivatives, integration with a firm exposition on series, sequences, and approximations are covered efficiently.

From the preface we learn that inspiration was to offer “Chinese students a course in English” and thus have a “suitable textbook” because “Chinese students are going overseas.” While it may also be true that this is “the first ever calculus textbook in China printed in color”, two things I expected from that preface are not found further in. I expected a Sino-centric delivery and possibly the awkward grammar found in many such translations. This, however, reads as if developed for the native English speaker from the onset. Also, the color that is present here is effectively and subtly applied pastel coloring to graphs and figures.

A graphical and numerical approach provides motivation and explanations for concepts around area, tangency, and measure approximation apt for the student beginning with a firm trigonometry and college algebra background. While barely over four hundred pages of main content is significantly less than similar texts covering the same ground, I find no gaps in coverage and opportunities to explain potentially confusing notation (interval versus point, etc.) are not forgone. The review of pre-requisite material, especially around sequences, goes well into the first two of the eight chapters, making the text self-contained. The use of sequences allows a good foundation on the topic of limits, including Cauchy’s Theorem, before the introduction of derivatives and integration.

The derivative is not introduced until Chapter 3, which contains excellent sections on “Related rates of change” and “The tangent line approximation and the differential”. Besides being ready classroom capsules themselves, they mark the end of about 170 pages of preface material so that a convenient starting point can be found for a wide variety of backgrounds. There are some calculus texts that start with integration — sensible from the standpoint of its longer history – but I feel that beginning with differentiation is best suited to most first-time calculus students. Differentiation here includes some parametric and polar forms, but these topics are really asides. More germane to the discussion, and marking it for applied study, is the focus on approximation from linearization to Taylor series. One particularly lucid section is “Curve sketching”, among the sections ready to deliver as a lecture or augment a less clear textbook.

As may be expected, the techniques for integration and improper integrals and applications of the definite integral as well as explorations of infinite series, sequences, and approximations tend toward introducing basic numerical methods such as Newton’s Method and Simpson’s Rule for the use of future physicists and software engineers. The final chapter on “Infinite series, sequences, and approximations” contains significant overlap to the beginning of the text, which is optional to some students, while extending naturally from the integral applications topic of the penultimate chapter. A concluding section to each chapter contains two or three score exercises without solutions. This is a concisely delivered introduction for engineering students and those on other applied tracks that could support two semesters of study.

Tom Schulte, a work at home software architect, spends significant recreation time bicycling paved and unpaved trails of Louisiana’s St. Tammany Parish.

See the table of contents in the publisher's webpage.