This book is associated with some rather large numbers:

- 14 main authors, of which the leading three are cited above.
- 132 other named contributors.
- 559 large pages and about 90 exercise sets, containing a total of approximately 3000 problems.
- 1500 answers, solutions and hints for the given problems.

Produced by the Calculus Consortium via an NSF grant, this second edition reflects critical feedback from teachers in over 200 schools in the USA, which accounts for some of the numbers quoted above. Also included in the preface are the main pedagogical principles used by the authors in the compilation of this tome. Here are the main ones:

- Stress on understanding and multiple ways of presenting mathematical concepts.
- Focus upon a small number of key concepts.
- Algebra review integrated throughout the text.
- Each function is represented symbolically, numerically, graphically and verbally.

However, the overarching educational principle is that mathematics is learned by doing, rather than reading or listening and this is reflected in the fact that the book consists mainly of problems to be solved. Some of these are routine practice exercises whilst others are more investigative.

Of course, no reviewer could/would check 3000 exercises for accuracy, so what I did was to take the 11th problem in each of the 90 sets and check for the clarity of the questions and accuracy of the solutions provided and not one mistake could I find. This level of precision also pertains to the main text, which is written in a lively manner and with an abundance of illustrations.

Calculus, since the time of Euler, has been a tool for the analysis of functions, so the main title of this book is therefore highly compatible with its subtitle. However, there is no coverage of approximate methods for the determination of the area included under a curve (e.g. the methods of Cavalieri, trapezoid rule etc) and this leads to the judgement that the book is really a preparation for the study of differential calculus.

It is stated that the book is intended for students who have successfully completed courses in intermediate algebra or high school algebra II and the functions under discussion are:

Linear, quadratic, exponential, logarithmic, trigonometric, polynomial and rational.

These are considered from the points of view of:

domain, range, function composition, inverse, concavity, asymptotic behaviour, periodicity, parameterisation, graphical representation etc.

With only a few exceptions, the underlying pedagogical aims are reflected in the development of the material and any students who complete even one third of the exercises will achieve good conceptual understanding in the realm of elementary functions. On the other hand, I feel that there should have been more explicit informal discussion on the notion of continuity, with greater emphasis paid to the difference between functions whose domains are discrete sets as opposed to whose domains are continuous subsets of the real numbers.

Chapter 7 provides a condensed introduction to complex numbers as extended solutions to quadratic equations. But this was given as Hobson's choice and at variance with pedagogical principle 1 (above). Complex numbers can, of course, be introduced in a variety of ways and, in particular, as sets of 2x2 matrices, which would have tied in nicely with the contents of chapter 10. They also have a geometric relevance, which received a negligible amount of discussion.

As for chapter 10, I found the treatment of vectors to be accurate, although highly conventional. This was followed by a rather standard introduction to matrices, the algebra of which, in my opinion, is treated rather instrumentally. Since these topics are not necessary precursors for an introduction to elementary calculus, their inclusion characterises a degree of content-overload.

In general, I think this book is a fine achievement. Not only does it provide firm foundations for the study of calculus but it will also serve as a mathematical handbook for ongoing study and the revision of basic algebraic skills. It has arisen from myriad of contributions and yet it exudes a continuity of style not usually associated with multiple authorship. For this reason alone, the main authors are to be congratulated.

Peter Ruane ([email protected]) is retired from university teaching, where his interests lay predominantly within the field of mathematics education.