Calculus Problems

The aim of this book is to provide a practical working tool for students in Engineering, Mathematics, and Physics, or in any other field where rigorous Calculus is needed. The emphasis is thus on problems that enhance students’ skill in solving standard exercises with a careful attitude, encouraging them to devote an attentive eye to what may or may not be done in manipulating formulae or deriving correct conclusions, while maintaining, whenever possible, a fresh approach, that is, seeking guiding ideas. Every chapter starts with a summary of the main results that should be kept in mind and used for the exercises of that chapter; this is followed by a selection of guided exercises. The theoretical preamble is meant to recapitulate the main definitions and results and should also offer a bird’s-eye view on the topic treated in the chapter. Hence, the student can quickly review the main theoretical facts and then, most importantly, “learn by examples,” becoming acquainted with the specific techniques by seeing them applied directly to the problems. Each exercise ends with a short comment which underlines the main issues of that specific exercise, the leading ideas, and the main techniques. A selection of problems closes each chapter, the answers to which are all listed in Solutions. The reader is urged to try to solve some of these problems, which are similar, but not always trivially analogous, to those that have been presented in detail.