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

✍ Scribed by Antonio Caminha Muniz Neto

• Publisher: Springer Verlag
• Year: 2017
• Tongue: English
• Leaves: 657
• Edition: 1st ed. 2017
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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