YEAR 4: Analytic Thinking

Logic.

• Propositional Logic. Formulas. Axiom Systems. Rules of Inference.

• Set Theory. Power of a Set. Axiom of Choice. Boolean Algebra.

• Induction. Descent. Contradiction.

• What is a Proof?

Concepts of Infinity.

• Convergence. Limits.

• Infinite Sequences and Series. Infinite Geometric Series.

• Natural Logarithm and Exponential. Continuous Compound Interest.

• Induction.

• Periodic Decimal Fractions.

• Archimedes Quadrature of the Parabola.

• Notion of Continuity.

Continued Fractions.

• Definition. Converting Continued Fractions and Simple Fractions.

• Nonterminating Continued Fractions.

• Expanding Functions into Continued Fractions

Trigonometry.

• Identities. Addition Formula. Product Formula.

• Solving Trigonometric Equations.

• Inverse Trigonometric Functions.

• Trigonometric Inequalities.

• Graphs of Trigonometric Functions.

• Law of Cosines. Law of Sines. Solving Oblique Triangles.

Geometry.

• Drawing and Construction Using Tools.

• Approximation of Areas and Volumes by Rectangles and Boxes.

• Loci of Points.

Approximation.

• Absolute and Relative Errors.

• Significant Digits. Rounding of Numbers.

• Error of a Sum, Product, Difference and Quotient.

• Relative Error of a Power or Root.

• Method of Bounds.

Inequalities and Means

• Inequality of Arithmetic and Geometric Mean

• Bernoulli Inequality. Mean Power Inequality. Holder Inequality.

• Use of Inequalities for Approximate Calculation.

• Cauchy-Schwarz Inequality.

• Quadratic Mean. Harmonic Mean.