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Advanced High-School Mathematics

(Tina Meador) #1

  • 1 Advanced Euclidean Geometry

    • 1.1 Role of Euclidean Geometry in High-School Mathematics

    • 1.2 Triangle Geometry

      • 1.2.1 Basic notations

      • 1.2.2 The Pythagorean theorem

      • 1.2.3 Similarity

        • theorems 1.2.4 “Sensed” magnitudes; The Ceva and Menelaus



      • 1.2.5 Consequences of the Ceva and Menelaus theorems

      • 1.2.6 Brief interlude: laws of sines and cosines

        • nius’ theorem 1.2.7 Algebraic results; Stewart’s theorem and Apollo-





    • 1.3 Circle Geometry

      • 1.3.1 Inscribed angles

      • 1.3.2 Steiner’s theorem and the power of a point

      • 1.3.3 Cyclic quadrilaterals and Ptolemy’s theorem



    • 1.4 Internal and External Divisions; the Harmonic Ratio

    • 1.5 The Nine-Point Circle

    • 1.6 Mass point geometry



  • 2 Discrete Mathematics

    • 2.1 Elementary Number Theory

      • 2.1.1 The division algorithm

      • 2.1.2 The linear Diophantine equationax+by=c

      • 2.1.3 The Chinese remainder theorem

      • 2.1.4 Primes and the 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 vi

      • 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





  • 3 Inequalities and Constrained Extrema

    • 3.1 A Representative Example

    • 3.2 Classical Unconditional Inequalities

    • 3.3 Jensen’s Inequality

    • 3.4 The H ̈older Inequality

    • 3.5 The Discriminant of a Quadratic

    • 3.6 The Discriminant of a Cubic

    • 3.7 The Discriminant (Optional Discussion)

      • 3.7.1 The resultant off(x) andg(x)

      • 3.7.2 The discriminant as a resultant

      • 3.7.3 A special class of trinomials





  • 4 Abstract Algebra

    • 4.1 Basics of Set Theory

      • 4.1.1 Elementary relationships

      • 4.1.2 Elementary operations on subsets of a given set

      • 4.1.3 Elementary constructions—new sets from old

      • 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

        • ation 4.2.2 Abstract algebra—the concept of a binary oper-



      • 4.2.3 Properties of binary operations

      • 4.2.4 The concept of a group

      • 4.2.5 Cyclic groups

      • 4.2.6 Subgroups

      • 4.2.7 Lagrange’s theorem vii

      • 4.2.8 Homomorphisms and isomorphisms

      • 4.2.9 Return to the motivation





  • 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 and l’Hˆopital’s rule



    • 5.2 Numerical Series

      • 5.2.1 Convergence/divergence of non-negative term series

      • 5.2.2 Tests for convergence of non-negative term series

        • ing series 5.2.3 Conditional and absolute convergence; alternat-

        • cussion) 5.2.4 The Dirichlet test for convergence (optional dis-





    • 5.3 The Concept of a Power Series

      • 5.3.1 Radius and interval of convergence

      • pansions 5.4 Polynomial Approximations; Maclaurin and Taylor Ex-

      • 5.4.1 Computations and tricks

      • 5.4.2 Error analysis and Taylor’s theorem



    • 5.5 Differential Equations

      • 5.5.1 Slope fields

      • 5.5.2 Separable and homogeneous first-order ODE

      • 5.5.3 Linear first-order ODE; integrating factors

      • 5.5.4 Euler’s method





  • 6 Inferential Statistics

    • 6.1 Discrete Random Variables

      • 6.1.1 Mean, variance, and their properties

      • 6.1.2 Weak law of large numbers (optional discussion)

      • 6.1.3 The random harmonic series (optional discussion)

      • 6.1.4 The geometric distribution

      • 6.1.5 The binomial distribution

      • 6.1.6 Generalizations of the geometric distribution

      • 6.1.7 The hypergeometric distribution viii

      • 6.1.8 The Poisson distribution



    • 6.2 Continuous Random Variables

      • 6.2.1 The normal distribution

      • 6.2.2 Densities and simulations

      • 6.2.3 The exponential distribution



    • 6.3 Parameters and Statistics

      • 6.3.1 Some theory

      • 6.3.2 Statistics: sample mean and variance

        • orem 6.3.3 The distribution ofXand the Central Limit The-





    • 6.4 Confidence Intervals for the Mean of a Population

      • lation variance 6.4.1 Confidence intervals for the mean; known popu-

      • ance 6.4.2 Confidence intervals for the mean; unknown vari-

      • 6.4.3 Confidence interval for a population proportion

      • 6.4.4 Sample size and margin of error



    • 6.5 Hypothesis Testing of Means and Proportions

      • 6.5.1 Hypothesis testing of the mean; known variance

      • 6.5.2 Hypothesis testing of the mean; unknown variance

      • 6.5.3 Hypothesis testing of a proportion

      • 6.5.4 Matched pairs



    • 6.6 χ^2 and Goodness of Fit

      • 6.6.1 χ^2 tests of independence; two-way tables





  • Index

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