INTRODUCTION TO MATHEMATICAL ANALYSIS

VIDEO LECTURES PRESENTED BY DR. MAU NAM NGUYEN

Introduction To Mathematical Analysis I, PDXOpen Textbooks, 2016

Chapter 1: Tools for Analysis

1.1 Basic Concepts of Set Theory

• • • Problem 1.1.2

1.2 Functions

• • • Problem 1.2.1

1.3 The Natural Numbers and Mathematical Induction

• • • Problem 1.3.1

    • Problem 1.3.7

1.4 Order Field Axioms

• • • Problem 1.4.5

1.5 The Completeness Axiom for the Real Numbers

• • • Upper Bounds and Lower Bounds

    • Supremum and Infimum: Part 1

    • Supremum and Infimum: Part 2

    • Supremum and Infimum: Part 3

    • Supremum and Infimum: Part 4

    • Problem 1.5.4

1.6 Applications of the Completeness Axiom

• • • Problem 1.6.3

Chapter 2: Sequences

2.1 Convergence

• • • Problem 2.1.10

    • Problem 2.1.13

2.2 Limit Theorems

• • • Problem 2.2.3

2.3 Monotone Sequences

• • • The Monotone Convergence Theorem: Part 1

    • The Monotone Convergence Theorem: Part 2

    • Problem 2.3.2

    • Problem 2.3.3

Chapter 3: Limits and Continuity

3.1 Limits of Functions

• • • Problem 3.1.6

3.2 Limit Theorems

• • • Problem 3.2.4

3.3 Continuity

• • • Problem 3.3.8

    • Problem 3.3.9

3.4 Properties of continuous functions

• • • The Extreme Value Theorem: Part 1

    • The Extreme Value Theorem: Part 2

    • The Extreme Value Theorem: Part 3

    • The Extreme Value Theorem: Part 4

    • The Intermediate Value Theorem: Part 1

    • The Intermediate Value Theorem: Part 2

    • The Intermediate Value Theorem: Part 3

    • Problem 3.4.5

    • Problem 3.4.6

3.5 Uniform Continuity

• • • Problem 3.5.2

    • Problem 3.5.3

Chapter 4: Differentiation

4.1 Definition and Basic Properties of the Derivative

• • • Problem 4.1.3

    • Problem 4.1.11

4.2 The Mean Value Theorem

• • • The Fermat Rule: Part 1

    • The Fermat Rule: Part 2

    • The Fermat Rule: Part 3

    • Problem 4.2.2

    • Problem 4.2.3

4.3 Some Applications of the Mean Value Theorem

• • • Problem 4.3.1

4.4 L’Hospital’s Rule

• • • Problems 4.4.5 & 4.4.6