Honors Calculus

Apparently following a national trend, obesity in calculus books was a problem for a while, but the book under review is positively anorexic. After praising the writing but criticizing the meager contents of Frank Morgan’s recent books (Real Analysis [RA], Real Analysis and Applications [RAA]), I am disappointed to find that Honors Calculus is even slimmer than Morgan’s progeny.

Intended either as a first-year honors course for the student who has taken calculus in high school or as a third-year analysis course, MacCluer’s book is very similar in spirit to Morgan’s volumes. MacCluer spends more time on sets and an axiomatic development of the real number system, but both authors develop topological notions (compactness, connectedness,…) early and use them to discuss limits and derivatives. There seems to be no mention of uniform convergence in MacCluer’s book, whereas Morgan gives the topic three pages. The writing style is correctly described by the author as “brief, breezy, and au courant — although the use of the word “wag” (meaning a person of wit) on p. 70 is jarring, as is the incorrect expression “by-in-large” used for “by and large” (derived from sailing) on p. 69. The term “basin of attraction” is not defined anywhere in the text, although the Index associates it with two exercises (in only one of which the term actually appears).

Honors Calculus claims to “survey the subject’s applicability to science and engineering.” This entails some brief discussion of interesting examples such as Kepler’s Laws, universal gravitation, statistics, and quantum mechanics. There is one example of a maximum-minimum problem (under “Optimization”) and one example of a related rate problem — a “beach-party problem” involving a lighthouse whose beam sweeps past the celebrants. There is a short section on differential equations with an example involving a simple first-order ODE . This leads into Kepler’s Laws and universal gravitation, with many of the details left as exercises. From time to time, pseudocode is given to aid in dealing with numerical aspects of a problem.

What really distinguishes MacCluer’s book from Morgan’s is the problem sets. [RA] and [RAA] give an average of 6-7 problems at the end of each section, many of them routine. MacCluer, on the other hand, accumulates problems at the end of his chapters, providing a rich stew of 25-50 exercises, routine (a few) and challenging intertwined. Among the interesting exercises is that of finding the maximum number of distinct sets that can be generated from any one set by repeated operations of closure and complementation (Chapter 3); the classic snowplow problem (Chapter 6); and the reason that the first several pages of a table of logarithms are more worn than the last several pages (Chapter 8). Many of the more challenging problems have hints or have their solutions outlined, although there are no answers/solutions at the back of the book (perhaps one aspect of the book’s being “in the European style”).

If I had a class of freshmen who had allegedly mastered basic calculus in high school, I might consider MacCluer’s book (anticipating a good deal of supplementing) — but I would be more inclined to choose Spivak’s Calculus, a marvelous book I have used to teach such an honors calculus course.

Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.