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Calculus of One Variable

Keith E. Hirst
Springer Verlag
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Hideo Nagahashi
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First, I was surprised how small the book is (9.0 by 6.8 inches, 240 pages).  Whenever I teach calculus using bigger textbooks, I feel guilty about letting students pay so much money for them and then covering only a part of the book in class.  What if there is a smaller text including exactly what I want to teach?

This is a textbook on single variable calculus which covers standard materials such as functions, limits (without the epsilon-delta definition), differentiation, Taylor expansions and integration.  Though the book is designed for beginning university students, it has some features (or defects) that make it valuable for only few readers.

The author has written the book as a transition between high school and university calculus courses in Europe, and so it requires readers of being already familiar with the subject to a certain extent.  For example, the derivatives of elementary functions are presented without justification, since the typical student would have learned those (as formal rules, perhaps) in their high school course.

On the other hand, the first third of the book (80 pages) reviews technical details of functions and limits which are not of primary importance, such as the secant, cosecant, cotangent, and even inverse hyperbolic functions and limits of functions of two variables (this in a book on single variable calculus!).

Though there are discussions on applications (one chapter each for differentiation and integration), the rest of the book seems to stress the computational aspects of calculus; important concepts are presented very briefly.  In particular, there are only a few pages of explanations on Riemann sums and the definite integral, followed by the mere statement of fundamental theorem at the beginning of Chapter 7. Integration is treated as antidifferentiation thereafter. The rest of Chapter 7 and Chapters 8, 9 and 10 (60 pages) are devoted for computing antiderivatives. 

A positive feature of the book is the extensive use of the Maple computer algebra system throughout. 

Overall, the author put into this little textbook too many of the technical and computational aspects of calculus, which are the ones I want to stress least.  If you have studied calculus before (at least a little), have a limited amount of time and money, want to learn computational aspects, especially those of integration and are able to use MAPLE, use this book.  But if you really want to learn what calculus is about, get a bigger, perhaps more expensive, but much better book.

Hideo Nagahashi is Visiting Assistant Professor of mathematics at Colby College in Waterville, ME.

 Functions and Graphs.- Limits of Functions.- Differentiation.- Techniques of Differentiation.- Applications of Differentiation.- Maclaurin and Taylor Expansions.- Integration.- Integration by Parts.- Integration by Substitution.- Integration of Rational Functions.- Geometrical Applications of Integration.- Answers to Selected Exercises.- Index.