Preface What happens when a student completes a course in “AP Calculus”, followed by a routine course in multivariable calculus, a computational course in linear algebra, and a formulaic presentation of differential equations? It is time for some real mathematics. There is still a world of interconnected labyrinths to explore, unscalable mountains to be climbed, and lands of mystery to be discovered. The first step in this is a rigorous course in one- variable analysis. This begins with a study of an ordered field in which the least upper bound property holds (not to be confused with a complete ordered field see the project in Section 2.7.3). Here the student meets Bolzano-Weierstrass, Heine-Borel, and a rigorous treatment of one-variable differentiation and integration with careful attention paid to the pervasive presence of the Mean Value Theorem. Then what? That is exactly the reason behind this book. Learning serious mathematics is about engaging with problems, from kindergarten to graduate school and beyond. The preliminaries for reading this book are already contained in the author’s book Tools of the Trade [27]. The first two chapters of Tools are Appendices A and B of this book. The reader who is familiar with that material can jump right into Chapter 1. The sequence of topics can be gleaned from the table of contents, so I will not dwell on that. There are three features here that should be discussed explicitly, espe- cially since they are important for the use of this book as a text. The first, and most important, is the collection of exercises. These are spread through- out the chapters and should be regarded as an essential component of the student’s learning. Some of these exercises comprise a routine follow-up to the material, while others will challenge the student’s understanding more deeply. The second feature is the set of independent projects presented at the end of each chapter. These projects supplement the content studied in their respective chapters. They can be used to expand the student’s knowl- edge and understanding or as an opportunity to conduct a seminar in Inquiry Based Learning (IBL) in which the students present the material to their class. A brief glance will show that the independent projects cover a wide range of interesting topics that hint at advanced areas of mathematics. The ix
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