Syllabus For The Montgomery County Mathematically Precocious Youth Program

Geometry

• Rules of Reasoning - modus ponens, modus tollens, deduction, double-denial, contrapositive, converses, biconditional sentences, Venn diagrams.

• Truth tables, Segments, midpoints, angles-complementary, supplementary, adjacent, vertical, Triangles - congruence theorems, perpendicular bisector, angle bisector, altitude, median, angle bisector, isosceles and equilateral triangles.

• Parallel lines, transversals.

• Polygons, similarity, necessary and sufficient conditions, Pythagorean Theorem, Trigonometric ratios, solving right triangles, solving oblique triangles.

• Coordinate Geometry - distance formula, midpoint formula, distance from a point to a line, slopes and perpendicularity.

• Circles - tangents, secants, chords, circumference, area. Areas of polygonal regions, areas of circular regions and sectors.

• Solid Geometry - points, planes and lines in space, volume, surface area, prismatoids, spheres, spherical triangles, locus.

Advanced Algebra

• The functional notation.

• Inverses, composition of functions.

• Exponents - integral, fractional, negative.

• Geometric Progressions, Factoring and the binomial theorem.

• Exponential functions, radicals, logarithms, linear interpolation.

• Logarithmic equations. Permutations and combinations.

• Mathematical induction.

• Continued sums and sigma notation, continued products.

Trigonometry

• Trigonometric functions - periodicity, odd and even functions, identities, graphing, amplitude, phase shift, graphing by the composition of ordinates.

• Special triangles, functions of 90, l80, 270 and multiples of 360.

• Further exploration of the Sine and Cosine Laws and the ambiquous case, problem-solving applications of Trigonometry.

• Addition and subtraction laws for the sine and cosine, product to sum formulas, double and Inverse Trigonometric functions - domain and range, graphing, principal-valued functions. Solving Trigonometric equations.

Analytical Geometry

• Conic sections - locus definitions and eccentricity.

• General second degree equation.

• Curve-sketching-asymptotes, symmetry, intercepts, excluded values, finding limits as the variable becomes infinite.

• Higher-order polynomials - synthetic division, equations involving radicals, equations in quadratic form.

• Imaginaries, the character and nature of the roots of a quadratic: forming an equation when roots are given, Descartes rule of signs.

• Graphing polynomials, multiplicity of roots.

• Polar coordinates, polar curves - limacons, rose-leaved curves, lemniscates, spirals, the conic sections in polar coordinates.

• Changing from rectangular to polar coordinates, the polar form of a complex number, modulus and argument, vectors, addition, subtraction, multiplication and division of complex numbers, DeMoivre's Theorem.