YEAR 4: Analytic Thinking
Learn concepts of infinity essential to the study of calculus and beyond. Approximation and errors to understand the limits of computer calculations. Learn logic, proofs and deduction.
Course Outline
Part I

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

Part II

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.

Part III

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.

CauchySchwarz Inequality.

Quadratic Mean. Harmonic Mean.
