An Excursion through Elementary Mathematics: Real Numbers and Functions: 1

Description

This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This first volume covers Real Numbers, Functions, Real Analysis, Systems of Equations, Limits and Derivatives, and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book. Preface Contents 1 The Set of Real Numbers 1.1 Arithmetic in R Problems: Section 1.1 1.2 The Order Relation in R Problems: Section 1.2 1.3 Completeness of the Real Number System Problems: Section 1.3 1.4 The Geometric Representation Problems: Section 1.4 2 Algebraic Identities, Equations and Systems 2.1 Algebraic Identities Problems: Section 2.1 2.2 The Modulus of a Real Number Problems: Section 2.2 2.3 A First Look at Polynomial Equations Problems: Section 2.3 2.4 Linear Systems and Elimination Problems: Section 2.4 2.5 Miscellaneous Problems: Section 2.5 3 Elementary Sequences 3.1 Progressions Problems: Section 3.1 3.2 Linear Recurrences of Orders 2 and 3 Problems: Section 3.2 3.3 The and Notations Problems: Section 3.3 4 Induction and the Binomial Formula 4.1 The Principle of Mathematical Induction Problems – Section 4.1 4.2 Binomial Numbers Problems – Section 4.2 4.3 The Binomial Formula Problems – Section 4.3 5 Elementary Inequalities 5.1 The AM-GM Inequality Problems: Section 5.1 5.2 Cauchy's Inequality Problems: Section 5.2 5.3 More on Inequalities Problems: Section 5.3 6 The Concept of Function 6.1 Definitions and Examples Problems: Section 6.1 6.2 Monotonicity, Extrema and Image Problems: Section 6.2 6.3 Composition of Functions Problems: Section 6.3 6.4 Inversion of Functions Problems: Section 6.4 6.5 Defining Functions Implicitly Problems: Section 6.5 6.6 Graphs of Functions Problems: Section 6.6 6.7 Trigonometric Functions Problems: Section 6.7 7 More on Real Numbers 7.1 Supremum and Infimum Problems: Section 7.1 7.2 Limits of Sequences Problems: Section 7.2 7.3 Kronecker's Lemma Problems: Section 7.3 7.4 Series of Real Numbers Problems: Section 7.4 8 Continuous Functions 8.1 The Concept of Continuity Problems: Section 8.1 8.2 Sequential Continuity Problems: Section 8.2 8.3 The Intermediate Value Theorem Problems: Section 8.3 9 Limits and Derivatives 9.1 Some Heuristics I 9.2 Limits of Functions Problems: Section 9.2 9.3 Basic Properties of Derivatives Problems: Section 9.3 9.4 Computing Derivatives Problems: Section 9.4 9.5 Rôlle's Theorem and Applications Problems: Section 9.5 9.6 The First Variation of a Function Problems: Section 9.6 9.7 The Second Variation of a Function Problems: Section 9.7 9.8 Sketching Graphs Problems: Section 9.8 10 Riemann's Integral 10.1 Some Heuristics II 10.2 The Concept of Integral Problems: Section 10.2 10.3 Riemann's Theorem and Some Remarks Problems: Section 10.3 10.4 Operating with Integrable Functions Problems: Section 10.4 10.5 The Fundamental Theorem of Calculus Problems: Section 10.5 10.6 The Change of Variables Formula Problems: Section 10.6 10.7 Logarithms and Exponentials Problems: Section 10.7 10.8 Miscellaneous Problems: Section 10.8 10.9 Improper Integration Problems: Section 10.9 10.10 Two Important Applications Problems: Section 10.10 11 Series of Functions 11.1 Taylor Series Problems: Section 11.1 11.2 Series of Functions Problems: Section 11.2 11.3 Power Series Problems: Section 11.3 11.4 Some Applications Problems: Section 11.4 11.5 A Glimpse on Analytic Functions Problems: Section 11.5 Bibliography Appendix A Glossary Appendix B Hints and Solutions Section 1.1 Section 1.2 Section 1.3 Section 2.1 Section 2.2 Section 2.3 Section 2.4 Sections 2.5 Section 3.1 Section 3.2 Sections 3.3 Section 4.1 Section 4.2 Section 4.3 Section 5.1 Section 5.2 Section 5.3 Section 6.1 Section 6.2 Section 6.3 Section 6.4 Section 6.5 Section 6.6 Section 6.7 Section 7.1 Section 7.2 Section 7.3 Section 7.4 Section 8.1 Section 8.2 Section 8.3 Section 9.2 Section 9.3 Section 9.4 Section 9.5 Section 9.6 Section 9.7 Section 9.8 Section 10.2 Section 10.3 Section 10.4 Section 10.5 Section 10.6 Section 10.7 Section 10.8 Section 10.9 Section 10.10 Section 11.1 Section 11.2 Section 11.3 Section 11.4 Section 11.5 Index