Review and outline books have long been popular supplementary materials for calculus students. In fact, I have often loaned my copy of Schaum's Outline of Calculus to students struggling with rules and applications of differentiation and integration. These books are especially popular among campus bookstore managers who keep their shelves well stocked waiting for midterms, when students decide their textbook is impossible to read — though up to this point they'd only opened their textbooks to copy down the homework sets. Shelling out a few bucks for a review book is much more convenient than going to class, finding their instructor's office during office hours, or actually reading the text from the beginning of the term, so calculus review books are big business.
Calculus takes the more traditional approach to calculus review, especially when compared to something like How to Ace Calculus. It includes the standard topics of single-variable calculus (with chapters titled Equations, functions, and graphs, Change, and the idea of derivative, The idea of limits, Computing some derivatives, Formulas for derivatives, Extreme values, the mean value theorem, and curve sketching, Word problems, The idea of the integral, Computing some integrals, Formulas for integrals, Geometric applications of the integral, and Motion), but differs from many other books of this type by providing a more thorough review of the concepts. The book certainly has its share of computational examples and problems, but Gootman also takes care to include fairly extensive "big picture" discussions and just enough review of necessary precalculus topics to keep from losing the reader — either from boredom or feeling lost. The index is put together nicely and the chapters and sections are well-labeled for easy access to specific information.
The examples and explanations are thorough and each chapter ends with a Summary of main points and a number of exercises, the solutions of which can be found in the appendix, following sections on trigonometric and logarithmic functions. Calculus has a thorough discussion of the definite integral, including applications such as volume (using the methods of "disks and washers" and "cylindrical shells"), initial value and motion problems, in addition to the "standard" derivative and antiderivative rules.
Although there are several applications sections, this book also takes a traditional view of calculus learning. For instance, there is no mention of calculator use beyond checking the book's calculations or integral approximation beyond the theoretical point of view that the definite integral of an integrable function is the limit of Riemann sums. I was also disappointed that the applications chapters were separate from the others and that one was called Word problems, but that's an editorial choice that is not uncommon.
Another criticism, albeit minor, is in response to some examples in the chapter on limits, where the reader is asked to compute the limit of (2x + 9)/(x – 3) as x approaches 4 "by evaluating the expression at x = 3.9, 3.99, and 3.999 and at x = 4.1, 4.01, and 4.001."
While I understand the point of such an exercise, I'm uncomfortable with the verb compute being used when approximate or estimate or guess seem more appropriate and serve to remind the reader that numerical evidence is useful, but often misleading.
Inexplicably, there is very little on antidifferentiation. Specifically, there is no mention of integration by parts, trigonometric substitutions, partial fractions, or use of tables of integrals. This is especially puzzling in light of the traditional nature of the book. I was unable to find a rationale, explicit or otherwise, for this glaring omission. In fact, it really feels like a chapter or two is missing. I strongly recommend that subsequent editions include this material.
I will close with a quote from the introduction, in which an anonymous student of Professor Gootman says "he could teach calculus to a cat." While I wouldn't go quite that far, I will say that he wrote a good exposition of calculus concepts and his book, despite the criticisms I've mentioned, could be a useful supplement for calculus students, feline and otherwise.
[S]Ayres, Frank & Mendelson, Elliott, Schaum's Outline of Calculus. Schaum's Outline Series, McGraw-Hill, 1990. ISBN 0070419736.
Steve Benson (firstname.lastname@example.org) is assistant professor at the University of Wisconsin–Oshkosh and Co-Director of the Master of Science for Teachers summer program at the University of New Hampshire.