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.
This well-written book will make a fine addition to the bookshelves of instructors and students alike. Its goals as similar to those of other such outlines, but it goes about it in a very different way. Compare these two paragraphs from the introduction to Calculus in the Barron's College Review series and the book under review:
If you are reading this introduction, then you probably want to learn more about calculus than you already know. Maybe you are the proverbial "intelligent layman" who just wants to get some idea of what calculus is all about. Maybe you had calculus many years ago and now find that you need a little bit of it for your work, but don't remember any.... But most of all this book is for high school and college students currently taking a course in calculus, and not doing so well at it. [Barron's]
If you are reading this introduction, this book is not for you. This book is directed at calculus students who have better things to do than read wordy preambles that won't be on the exam. [How to Ace]
How to Ace Calculus takes a tongue-in-cheek approach and includes "the lowdown" on non-mathematical topics such as choosing and dealing with your instructor, asking questions, preparing for and taking exams. It also does some mathematics — providing entertaining takes on the standard concepts of the first year of calculus, providing (often bizarre) problems which illustrate or apply these concepts, and includes a useful final summary section, "Just the Facts: Quick Reference Guide."
While it occasionally tries a little too hard to be funny, it does provide an edifying look behind the lecturer's podium and will provide many students with useful suggestions. I believe, however, that it might be even more popular among instructors — especially those who don't take themselves too seriously — who will nod in appreciation at familiar situations. While I was sometimes put off by the overblown caricatures of mathematicians and mathematics departments, I have to admit that I often laughed out loud when "recognizing" people or situations being made fun of. I am certain that some instructors, especially those who would prefer not to be teaching (either calculus or at all), will not like the book, since the authors provide an often brutally honest look at faculty at institutions where teaching is either much less-valued than research or not valued at all.
The first part of the book consists of five short "advice" chapters, mostly organized into lists, Exactly who and what is your instructor, General principles of acing calculus, Good and bad questions, Are you ready?, Calc preps, and How to handle the exam. Much of the advice provided is very useful, while some is exaggerated for the purposes of entertainment. For instance, the book gets off to a rollicking start in Exactly who and what is your instructor, where the authors provide clues based on the instructor's title:
The list goes on to include graduate students and those faculty who have no signs (or no doors), but the authors do go into more detail to explain that good and bad teachers occur in all categories and that their characterizations are "wild generalizations." In the same chapter, the book gives some common sense advice on how to find the best instructor, what to expect from an instructor (including some funny — if apocryphal — anecdotes about Von Neumann, Wiener, and Hilbert), and how to deal with an instructor.
In the next chapters, the authors continue to give good advice, including recommendations about doing homework, getting help, studying for exams, and taking exams. The chapter on calculus prerequisites provides a very brief review of some essential ideas. As this is not a text, no exercises — and very few examples — are given, but some key concepts are reviewed very briefly: functional notation and composition, absolute value, geometry, and trigonometry, and a short section on the use of calculators.
The next 22 chapters (remember, they're all fairly short) focus on various calculus topics. While it is entertaining and educational (consisting of examples involving plastic spotted owls, dropping calculus books off the Golden Gate Bridge, tracking UFOs, and peanut butter guacamole dip rumored to be an aphrodisiac, to name a few), much of the mathematical information is very similar to the exposition in many calculus (especially reform-based) texts. For instance, Calculus from Graphical, Numerical, and Symbolic Points of View by Ostebee and Zorn [OZ] also does an especially nice job of combining humor and well-written mathematics. That being said, the explanations here are well done, and while there aren't a lot of examples, they are well chosen.
I think my favorite chapter might be Twenty most common exam mistakes and not just because the authors take a cue from The Big Ten athletic conference and include 21 items in their list. If students take the time to read these warnings and take them to heart, they will be spared a lot of pain when they get their tests back.
I did find a few instances where I felt the authors failed to make important points, though. The book seems to be aimed at students taking calculus at a research university in large lecture format using a traditional text, and fails to to acknowledge the existence of texts in which the problem sets are not all but identical to the examples.And at times opportunities are missed. For example, after providing a calculator program designed to provide evidence that the limit of sin(x)/x as x tends to 0 is equal to 1 (by looking at sin(101–k )/101–k for k = 1, 2, ... 8), they fail to point out that this does not always work — e.g. replace sin(x) with sin(107πx). Now, I'm not lobbying for deltas and epsilons, but I feel they missed an opportunity to make an important point about limits (it's sometimes hard to tell if you're "close enough" to guess the limit).
Similarly, in a later chapter, they talk about the antiderivative rather than taking the opportunity to remind the reader that any function having an antiderivative actually has infinitely many antiderivatives. Finally, there was very little information or examples concerning applications of the definite integral other than for computing the area under a curve. I consider these fairly minor omissions, though, as this is a supplement, not a textbook.
Although I have no genealogical evidence to back it up, I'm convinced that one of the authors must be related to Douglas Adams, the author of the five(!) books in the Hitchhiker's Guide to the Galaxy trilogy. All that was missing was DON'T PANIC appearing in large block letters (of course, I want credit when this appears in the next edition!). Although Professor Adams seems to be the most likely "suspect" (based on his name and my experience with one of his alter egos, Mel Slugbait), any or all of the authors could be responsible for the overall humorous tone. The glossary is especially entertaining (I'll leave it to the reader to discover the many gems) and even helpful at times, more evidence that the authors have done a good job of balancing humor with useful information.
I'll close my review of How to Ace Calculus with the authors' own words, found at the end of the introduction:
If you approach it with the right point of view, learning calculus can be not only a mind-expanding experience but also fantastic fun, just about as good as something not involving whipped cream and maraschino cherries can get. This book is going to tell you how calculus is taught, how to get the best teachers, what to study, and what is likely to be on exams. This is the stuff we wish we'd known when we had to take calculus. So, enough stalling. Why don't you go up to that nice cashier, plunk down some money and buy the book, and we can talk more later?
[B] Gootman, Elliot C., Calculus. Barron's College Review Series, 1997. ISBN: 0-8120-9819-6.
[S] Ayres, Frank & Mendelson, Elliott, Schaum's Outline of Calculus. Schaum's Outline Series, McGraw-Hill, 1990. ISBN 0070419736.
[OZ] Ostebee, Arnie & Zorn, Paul, Calculus: from Graphical, Numerical, and Symbolic Points of View. Saunders College Publishers, 1997. ISBN 0-03-019587-X.
Steve Benson (email@example.com) 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.