Bøger af Andrei Bourchtein

This textbook focuses on the study of different kinds of elementary functions ubiquitous both in high school Algebra and Calculus. To analyze the functions ranging from polynomial to trigonometric ones, it uses rudimentary techniques available to high school students, and at the same time follows the mathematical rigor appropriate for university level courses.Contrary to other books of Pre-Calculus, this textbook emphasizes the study of elementary functions with rigor appropriate for university level courses in mathematics, although the exposition is confined to the pre-limit topics and techniques. This makes the book useful, on the one hand, as an introduction to mathematical reasoning and methods of proofs in mathematical analysis, and on the other hand, as a preparatory course on the properties of different kinds of elementary functions.The textbook is aimed at university freshmen and high-school students interested in learning strict mathematical reasoning and in preparing a solid base for subsequent study of elementary functions at advanced level of Calculus and Analysis. The required prerequisites correspond to the level of the high school Algebra. All the preliminary concepts and results related to the elementary functions are covered in the initial part of the text. This makes the textbook suitable for both classroom use and self-study.

This book presents counter examples to false statements typically found within the study of mathematical analysis, real analysis, and calculus, all of which are related to infinite sequences, series of functions, and functions and integrals depending on a parameter.

This book provides a one-semester undergraduate introduction to counterexamples in calculus and analysis. It helps engineering, natural sciences, and mathematics students tackle commonly made erroneous conjectures. The book encourages students to think critically and analytically, and helps to reveal common errors in many examples. It is intended for students who are interested in developing a deeper understanding of the topics of calculus. The gathered counterexamples may also be used by calculus instructors in their classes.