Advanced High-School Mathematics

✍ Scribed by David B. Surowski

• Year: 2011
• Tongue: English
• Leaves: 435
• Category: Library

✦ Table of Contents

further
Preface
Contents
Chapter 1: Advanced Euclidean Geometry
1.1 Role of Euclidean Geometry
1.2 Triangle Geometry
1.2.1 Basic notations
1.2.2 The Pythagorean theorem
1.2.3 Similarity
1.2.4 Ceva and Menelaus
1.2.5 Consequences
1.2.6 Laws of sines and cosines
1.2.7 Stewart and Apollonius
1.3 Circle Geometry
1.3.1 Inscribed angles
1.3.2 Steiner's theorem; power of a point
1.2.3 Cyclic quadrilaterals and Ptolemy's theorem
1.4 Harmonic Ratio
1.5 Nine-Point Circle
1.6 Mass Point Geometry
Chapter 2: Discrete Mathematics
2.1 Elementary Number Theory
2.1.1 The division algorithm
2.1.2 Linear Diophantine equations
2.1.3 Chinese Remainder Theorem
2.1.4 Fundamental theorem of arithmetic
2.1.5. The principle of mathematical induction
2.1.6 Fermat's and Euler's theorems
2.1.7 Linear congruences
2.1.8 Alternative number bases
2.1.9 Linear recurrence relations
2.2 Elementary Graph Theory
2.2.1 Eulerian trails and circuits
2.2.2 Hamiltonian cycles and optimization
2.2.3 Networks and spanning trees
2.2.4 Planar graphs
Chapter 3: Inequalities and Constrained Extrema
3.1 A Representative Example
3.2 Classical Unconditional Inequalities
3.3 Jensen's Inequality
3.4 The Hoelder Inequality
3.5 The Discriminant of a Quadratic
3.6 The Discriminant of a Cubic
3.7 The Discriminant
3.7.1 The resultant
3.7.2 The discriminant as a resultant
3.7.3 Special class of trinomials
Chapter 4: Abstract Algebra
4.1 Basics of Set Theory
4.1.1 Elementary relationships
4.1.2 Elementary operations
4.1.3 Elementary constructions
4.1.4 Mappings between sets
4.1.5 Relations and equivalence relations
4.2 Basics of Group Theory
4.2.1 Motivation---graph automorphisms
4.2.2 Concept of a binary operation
4.2.3 Properties of binary operations
4.2.4 Concept of a group
4.2.5 Cyclic groups
4.2.6 Subgroups
4.2.7 Lagrange's theorem
4.2.8 Homomorphisms and isomorphisms
4.2.9 Return to the motivation
Chapter 5: Series and Differential Equations
5.1 Quick Survey of Limits
5.1.1 Basic definitions
5.1.2 Improper integrals
5.1.3 Indeterminate forms
5.2 Numerical Series
5.2.1 Convergence/divergence
5.2.2 Tests for convergence
5.2.3 Conditional vs. absolute convergence
5.2.4 The Dirichlet test for convergence
5.3 The Concept of a Power Series
5.3.1 Radius of convergence
5.4 Maclaurin and Taylor Expansions
5.4.1 Computations and tricks
5.4.2 Taylor's theorem
5.5 Differential Equations
5.5.1 Slope fields
5.5.2 Separable and homogeneous ODE
5.5.3 Integrating factors
5.5.4 Euler's method
Chapter 6: Inferential Statistics
6.1 Discrete Random Variables
6.1.1 Mean, variance, and their properties
6.1.2 Weak law of large numbers
6.1.3 The random harmonic series
6.1.4 Geometric distribution
6.1.5 Binomial distribution
6.1.6 Generalizations of the geometric distribution
6.1.7 Hypergeometric distribution
6.1.8 Poisson distribution
6.2 Continuous Random Variables
6.2.1 Normal distribution
6.2.2 Densities and simulations
6.2.3 Exponential distribution
6.3 Parameters and Statistics
6.3.1 Some theory
6.3.2 Statistics: sample mean and variance
6.3.3 Central Limit Theorem
6.4 Confidence Intervals for the Mean
6.4.1 Known population variance
6.4.2 Unknown variance
6.4.3 For proportions
6.4.4 Margin of error
6.5 Hypothesis Testing of Means and Proportions
6.5.1 Known variance
6.5.2 Unknown variance
5.5.3 For proportions
5.5.4 Matched pairs
6.6 Goodness of Fit
5.6.1 Independence and two-way tables
Table of Critical t-values
Index