ADVANCED ALGEBRA SYLLABUS
Real numbers system Review
Functions: i. Domain and range; evaluations; increasing and decreasing; interval notation; even and odd; vertical line test; transformations (including reflections, translations and dilations); basic graphs; combinations (sum, difference, product, quotient and composition); inverse; horizontal line test; ii. Linear review - equations; properties of slopes iii. Quadratic review - equations; factors; formula; discriminant; completing the square; standard form for transformations. iv. Polynomials of degree greater than 2 - graphs; equations and solutions (zeros or intercepts); synthetic division; remainder theorem; factor theorem; rational zero theorem; Descartes rule of signs; bounds for roots. v. Rational functions - domain and range; graphs; asymptote criteria (vertical, horizontal and slant); symmetry; vi. Exponential and Logarithmic - domain and range; graphs; solving equations; properties; different bases; simplifying; natural exponential and natural logarithmic functions; applications involving growth and decay.
Polar coordinate system: trigonometric functions and the unit circle; radian measure; periodicity; conversions from rectangular to polar and polar to rectangular in real numbers; graphs.
Complex numbers: operations in a + bi form; conjugates; modulus; geometric representations; polar form; conversions from rectangular to polar and polar to rectangular form;multiplication and division in polar form; powers and roots (De Moivre's theorem); roots of unity.
Additional topics; mathematical induction; conic sections (standard equations); finite sums and series ( including arithmetic and geometric sums and sums to infinity); review of binomial expansion.
INTRODUCTION TO CALCULUS SYLLABUS
The Derivative: Slope of curved lines; general limits; tangents to a curve at a point; limit (as h approaches 0) of [f(x + h) - f(x)]/h; rules of differentiation (power, product,quotient and chain); simple trigonometric derivatives.
Applications of the Derivative: including curve sketching; maxima and minima; first and second derivative test; second derivative; concavity, points of inflection; tangent and normal; distance and velocity problems; word problems (including simultaneous equations involving derivatives); related rates.
The Integral: Riemann sums and the definite integral; The Fundamental Theorems of Calculus; rules for integration; simple trigonometric integrals
Applications of the Integral: including simple differential equations (general and particular solutions); areas of plane regions - enclosed by curve and x axis and enclosed between two curves; volumes of solids of revolution (rotations about the x axis); accelerated motion, velocity and distance problems.