This course is designed for the most successful mathematics students who either have a genuine interest in mathematics and enjoy meeting its challenges and problems, or need such mathematics for further studies or related subjects such as physics, engineering, and technology at university level.  Students will study a wide range of complex topics in depth including vectors, matrices, coordinate geometry, trigonometry, probability, statistics, differential and integral calculus, abstract algebra, and review for the IB exam.  The prerequisites for this course are a minimum of an A average in Algebra II/Trigonometry and teacher recommendation.

Learning Objectives
    The student will:
        Demonstrate the following skills
 

Fundamental Mathematics

Sum and Product of Roots of Quadratic Equation
    The student will:
        Find roots of quadratic equations using sum and product rules for roots.

Factor and Remainder Theorems
    The student will:
        Find solutions to polynomial equations using factor and remainder theorems.

Partial Fractions
    The student will:
        Split rational functions with repeated and quadratic roots into partial fractions.
        Use partial fractions to assist in finding derivatives and integrals.

Inequalities in One Variable
    The student will:
        Graph and solve inequalities in one variable.

Radical Equations
    The student will:
        Graph and solve equations involving radical expressions.

Pairs of Equations
    The student will:
        Solve both linear and nonlinear simultaneous equations.

Arithmetic and geometric series
    The student will:
        Use formulas to find specific terms, sums, and sums to infinity.

Mathematical Induction
    The student will:
        Do mathematical proof by induction.

Permutations and Combinations
    The student will:
        Calculate the number of ways of selecting and arranging objects.

Binomial Theorem
    The student will:
        Find specific terms of binomial expansions with positive integer indices.

Sine and cosine rules
    The student will:
        Find solutions in triangles in both 2-d and 3-d.

Trigonometrical Identities
    The student will:
        Use trigonometry identities to solve trigonometrical equations.

Trigonometrical Equations
    The student will:
        Use acosx + bsinx = Rcos(x-alpha).
        Solve trigonometrical equations and find multiple solutions within a given range.

Complex Numbers
    The student will:
        Use Euler's formula.
        Use the four basic operations on complex numbers.
        Use complex numbers to find roots of equations.
        Use conjugate roots to simplify and solve equations.
        Use De Moivre's Theorem to solve equations.

Probability

Finite and Infinite
    The student will:
        Apply the basic concepts of probability.

Tree and Venn Diagrams
    The student will:
        Use tree diagrams and Venn diagrams to solve problems involving probability.

Independence
    The student will:
        Use the formula for independence to solve problems involving  probability.

Conditional Probability
    The student will:
        Use the formula for conditional probability.
        Construct tree diagrams to illustrate conditional probability.
        Use Bayes' Theorem to solve problems involving conditional probability.

Discrete Probability Distributions
    The student will be able to find and use:
        Expectation
        Mode
        Median
        Variance
        Standard deviation
        Calculate the expectation, mode, median, variance, and standard deviation for discrete functions.

Binomial Distribution
    The student will:
        Calculate the mean, variance, and standard deviation of a binomial distribution.

Hypergeometric Distribution
    The student will:
        Calculate the probability of a series of events without replacement.

Continuous Probability Distributions
    The student will find and use:
        Expectation
        Mode
        Median
        Variance
        Standard deviation
        Calculate the expectation, mode, median, variance, and standard deviation for continuous functions.

Normal Distribution
    The student will:
        Use the Normal distribution to find probabilities, mean, variance, and standard deviation.
        Use the Normal distribution table to find values.
        Use linear interpolation.

Normal Approximation
    The student will:
        Create and use the Normal Approximation to the Binomial Distribution.
        Use continuity corrections.

Functions and Calculus

Function, Domain, Codomain, and Range
    The student will:
        Use domain and range to graph functions and identify where a function is undefined.
        Given the graph of a function, state the domain and range.

Graphs of Elementary Functions
    The student will:
        Graph related functions given the original function.
        Graph translations, absolute values, and inverses.

Greatest and Least Values
    The student will:
        Find the greatest and least values of a function on a given interval.

Continuity
    The student will:
        Identify continuous and discontinuous functions.

Periodic, Even and Odd, Inverse, Reciprocal Functions
    The student will:
        Identify and graph periodic, odd, even, and reciprocal functions.
        Locate asymptotes, local maxima, and local minima.

Polynomial and Rational Functions
    The student will:
        Graph and solve polynomial and rational functions.

Composite and Absolute Value Functions
    The student will:
        Graph and solve both composite and absolute value functions.

Exponential and Logarithmic Functions
    The student will:
        Graph and solve both exponential and logarithmic functions.

Trigonometric Functions and Inverse
    The student will:
        Identify both domain and range for each function and its inverse.
        Graph and solve trigonometric functions and their inverses.

Derivative

Graphical interpretation
    The student will:
        Find simple derivatives using the definition of derivative.
        Sketch a graph of the derivative given the original function.

Approximation
    The student will:
        Use the approximation formula to estimate f'(x + h).

Derivatives
    The student will find derivatives of the following types of functions:
        Algebraic
        Logarithmic
        Exponential
        Trigonometrical
        Inverses

Sums, Products and Quotients
    The student will:
        Find the derivative for sums, products, and quotients using sum, product, and quotient rules.

Chain Rule
   The student will:
        Find the derivative of composite functions using the chain rule.

Second Order Derivatives
    The student will:
        Find the second derivative given the original function.

Implicit or Parametric Derivatives
    The student will:
        Find the derivative of functions given implicitly or parametrically.

Related Rates of Change
    The student will:
        Solve real world applications using related rates.
        Use geometrical formulas of area, volume, and pythagorus.

Tangents and Normals
    The student will:
        Find equations for both tangents and normals given a function and a point on the function.

Applications of Differentiation .
    The student will:
        Use derivatives to solve equations involving displacement, velocity, and acceleration.

Stationary Points
    The student will:
        Find all maximum, minimum, and inflection points.

Curve Sketching
    The student will:
        Sketch graphs of curves showing all intercepts, asymptotes, and stationary points.

Definite and Indefinite Integrals
    The student will:
        Use the Fundamental Theorem of Calculus.

Techniques of Integration
    The student will use the following techniques:
        Substitution
        Integration
        Reductions formulae
        Arctan and arcsin to solve related integrals.

Use of Integration
    The student will:
        Find the area of bounded regions
        Find the volume of a rotation using integration.

Applications of Integration
    The student will:
        Given the derivative function, find the original function through a given point.

Verification of Function
    The student will:
        Determine whether the function satisfies the differential equation.

Representation of a Situation
    The student will:
        Use differential equations to solve growth and decay problems.
        Use separation of variables to solve differential equations.
        Find and use the correct integrating factor to solve differential equations.

Matrices and Vectors

Matrices
Element
    The student will:
        Identify each of these terms in a given matrix.
            Row
            Column
            Dimension

Algebra of Matrices
        Equality
        Addition
        Subtraction and multiplication by a scalar
        Multiplication of two matrices
        Use matrix algebra to combine matrices.

Determinants of Matrices
    The student will:
        Find the determinant for any n x n matrix.
        Identify singular matrices.

Inverse of an n x n matrix
    The student will:
        Use the reversal rule (AB)^-1 = B^-1 A^-1
        Use the reversal rule to solve matrix equations.

Solution of Linear Equations
    The student will understand the following cases:
        Unique solution
        No solution
        Infinity of solutions
    The student will:
        Find solution(s), if any exist, of a set of linear equations using matrix techniques.
        Write solution(s) as a point, line, or plane.

Vectors in Two and Three Dimensions
    The student will:
        Graph vectors in 2 and 3 dimensions.
        Represent vectors using vector notation.

Components of a Vector
    The student will:
        Identify the components of a vector.

Addition and Multiplication of Vectors
    The student will:
        Add vectors.
        Multiply vectors by a scalar.
        Graph the resultant vector.

Length of Vector
    The student will:
        Calculate the length (magnitude) of a vector.
        Write down the unit vector for a given vector.

Scalar Products and Projections
    The student will:
        Calculate the scalar product.
        Find the angle between 2 vectors using scalar product formula.
        Prove orthogonality.

Vector Products
    The student will:
        Calculate the vector product of 2 vectors.
        Find a 3rd vector that is perpendicular to 2 other vectors.

Geometric Application of the Vector Product
    The student will:
        Use of the formula involving vector product, magnitudes, and sine of the angle.

Algebra of Scalar and Vector Products .
    The student will:
        Show that two vectors are orthogonal.
        Use the formula magnitude of cross product equals the product of the squared magnitudes minus the square of the dot product.

Lines and Planes
    The student will:
        Express equations of lines and planes in parametric, cartesian, and normal form.

Intersections
    The student will find the intersection of:
        Two lines
        Line with a plane
        Two planes
        Three planes

Distances in Three Dimensions
    The student will:
        Write expressions and calculate the distance between 2 objects (points, lines, or planes) in 3-d.

Abstract Algebra
 

Review for the IB Exam to be given in May.