This is a nice book for a student who has just had an introduction to calculus and wants to get deeper into the subject. Going through the entire book will not be easy, but it will certainly be very rewarding. Most of the material is presented as a series of problems, which students can explore at their own pace. There are many interesting problems and unusual results. The book can be used for self-study for motivated and well-prepared students, but most students will probably enjoy it more if they can benefit from instructor guidance.

As the title suggests, the theme of the book is that calculus is a powerful tool for computing approximations. This theme is explored in many concrete problems, with especial emphasis on how to approximate π(x), the number of prime numbers between 1 and x. The author convinces the reader that π can be well approximated by reasonably simple, differentiable functions, even though it is not a differentiable function.

The book is very well written and contains many references to articles in journals that are accessible to students, such as the *American Mathematical Monthly,* *Mathematics Magazine*, and *The College Mathematics Journal*. Looking up those references will be a good introduction to the mathematical literature.

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.