You are here

Approximately Calculus

Shahriar Shahriari
American Mathematical Society
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Luiz Henrique de Figueiredo
, on

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.

  • Patterns and induction
  • Divisibility
  • Primes
  • Derivatives and approximations of functions
  • Antiderivatives and integration
  • Distribution of primes
  • Log, exponential, and the inverse trigonometric functions
  • The mean value theorem and approximations
  • Linearization topics
  • Defining integrals, areas, and arclengths
  • Improper integrals and techniques of integration
  • The prime number theorem
  • Local approximation of functions and integral estimations
  • Sequences and series
  • Power series and Taylor series
  • More on series
  • Limits of functions
  • Differential equations
  • Logical arguments
  • Hints for selected problems
  • Bibliography
  • Index