Preface

Calculus is an essential mathematical tool for physical and natural phenomena analysis; it is a very important subject for students in the Faculties of Science and Engineering. This ebook has been designed to cover one academic semester for first-year students, as it contains the fundamentals related to this discipline. Here, the reader will find a concise and clear study of this mathematical field, it provides many examples, exercises, and case studies in each chapter. The complete solutions to the exercises are also included at the end of the ebook. The reader will be able to understand it, even without rigorous mathematical knowledge and will be able to immediately practice the concepts. Its main purpose is to enable students to put into practice what they are learning about the subject.

This ebook thoroughly examines the algebraic structure of the field and characterizes the set of real numbers, as a group formed with the natural, integer, rational, and irrational numbers. It introduces the family of functions, their geometric representation, properties, and restrictions. It reviews the concept of map in order to show its potential advantages over the functions. The limit operator, an essential operator in this discipline, is defined and some examples are given. This operator has a major role in the definition of continuity and differentiability of a function. With these topics, the characterization of this family of functions is concluded.

With the concept of limits, the sequences and series are introduced and the techniques to identify their convergence are reviewed. Finally the composition of functions, their implicit differentiation, as well as Fourier series, and Taylor series are studied.

The author hopes the reader interested in the study of the fundamentals of differential and integral calculus, finds useful the material presented here and that those who start studying this field find this information motivating. The author would like to acknowledge the Faculty of Sciences at Universidad Nacional Autónoma de México for support.