This is a textbook about Calculus of One Variable for students of Mathematics in the 21 st century. Its objective is to really teach students, i.e. to provide them with the necessary knowledge about Calculus of a One Variable and build up their abilities to use this wealth of information constructively in applications. In calculus, this means that the students should learn the concepts, the principles, the theorems, and the properties that relate the various mathematical entities, they should solve the exercises and the problems and they should improve their ability to apply their new knowledge in applications. A calculator of any type (for numerical or algebraic calculations) must be considered as a helping servant who cannot and must not be allowed to replace a thinking mind and a writing hand.
This course is designed to cover a little more than the needs of an average first year student in mathematics, physics or engineering. It consists of eight chapters:
Chapter 1: Numbers
Chapter 2: Sequences
Chapter 3: Functions
Chapter 4: Limits and Continuity
Chapter 5: Derivatives
Chapter 6: Integrals
Chapter 7: Series
Chapter 8: Generalized Integrals
Each chapter contains the basic theory and many related problems. The theory is supplemented with material given online in various forms (examples, theorems, proofs, calculations, extensions, applications, etc.). Also, the solutions of the problems are given online. Thus the printed part of the book remains small and constitutes the basis for building the course. The online material can be easily reached and is used for two purposes: to enhance the amount of knowledge provided to the student and to facilitate the solution of the problems. The student should read and understand the printed part, read and try to understand the material of the additions and then try to solve the problems. The most appropriate method for learning calculus is to read carefully the theory and try to use it by solving problems and in applications.
This book has been created with great respect to the student of the 21st century, in order to support their strenuous efforts for education in limited time and financial means. 1 have tried to make it easier to understand and to use, thorough and rigorous, with many inovations. But most of all I did it My Way.
(from the preface of Sotirios C. Persidis)