Many people have some experience with calculus. It is known that techniques from calculus are the basis of some important branches of mathematics, including analysis and analytic number theory. Indeed, it was the careful study of the concepts underlying the calculus that sowed the seeds of analysis. Euler’s intelligent proof of the divergence of the series of the reciprocals of the primes used only (delicate) ideas from calculus. His proof helped lead Dirichlet and Riemann to lay down the foundation stones of analytic number theory.

Usually, when we teach a course in calculus, we teach a few key topics and subjects and, if we are lucky, we will be able to solve some practical examples for students. The result of this is that we are forced to omit many beautiful ideas. Probably aware of this, many calculus books leave these brilliant results to the final exercises of their chapters. This sad gap becomes more annoying when we remember that the historical development of the calculus was largely spurred by the discovery of just those results that we usually disregard.

The book under review is a remarkable resource to fill this gap. Although this book is intended as a second course in calculus of a single variable, it follows the standard structure of calculus books. The author gives beautiful proofs of some of the basic results and focuses on deducing from them several interesting facts.

A large number of inequalities are proved, including numerical, geometrical, and differential and integral inequalities. The author presents several proofs of the arithmetic-geometric mean inequality, studies other possible means of numbers, and gathers some inequalities concerning the number \(e\). He discusses various mean value theorems and considers the concept of convexity and its applications.

Integrals and methods to compute them are explained as well. Several brilliant classical examples of integration are given in detail, which includes studying Wallis’s product for \(pi\), proving irrationality of some famous numbers, Euler’s sum of the reciprocals of the squares, the divergence of the series of reciprocals of the prime numbers, and Stirling’s formula for factorials.

The author has gathered several interesting exercises at the end of each chapter. Many of exercises are selected from educational notes, and give research ideas at the undergraduate level for readers. Each chapter ends with a rich list of references.

The book is very well motivated and well written, and is very suitable for readers to follow as self-study. I recommend it highly for undergraduate students who have completed a course in calculus, and also for students who have a course in mathematical analysis. It will also serve well for anyone who is interested in developing undergraduate research at the calculus level.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.