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      A) Arithmetic and Basic Algebra

•        Understand and work with rational, irrational, real, and/or complex numbers. Use numbers in a way that is most appropriate in the context of a problem. (Eg., appropriately rounded numbers, numbers written in scientific notation, using 5(100-1) for 5(99),etc.).
•         Demonstrate an understanding of the properties of counting numbers. (E.g., prime or composite, even or odd, factors, multiples).
•          Apply the order of operations to problems involving addition, subtraction, multiplication, division, roots, and powers, with and without grouping symbols.
• Identify the properties (e.g., closure, commutativity, associativity, distributivity) of the basic operations (i.e., addition, subtraction, multiplication, division) on standard number systems
• Given newly defined operations on a number system, determine whether the closure, commutative, associative, or distributive properties hold.
• Identify the additive and multiplicative inverses of a number.
• Interpret and apply the concepts of ratio, proportion, and percent in appropriate situations.
• Solve problems that involve measurements in the metric or the traditional system.
• Solve problems involving average, including arithmetic mean and weighted mean.
• work with algebraic exoressions and formulas. (E.g., If x = 5 and y = 6, what is the value of.

          ,then express h

•        Add, subtract, multiply, and divide polynomials, as well as algebraic fractions.
• Translate verbal expressions and relationships into algebraic expressions or equations.
• Solve and graph linear equations and inequalities in one or two variables; solve and graph systems of linear equations and graph inequalities in two variables: solve and graph nonlinear algebraic equations and graph inequalities.
• Determine any term of a binomial expansion using pascal's triangle or some other method. (E.g.,              )
• Present geometric interpretations of algebraic principles. (E.g, the triangle inequality and the distributive principles)

       B)   Geometry

•         Solve problems involving the properties of parallel and perpendicular lines.
• Solve problems using special triangles, such as isosceles and equilateral
• Solve problems using the relationships of the parts of triangles, such as sides, angles, medians, midpoints, and altitudes.
• Apply the Pythagorean theorem to solve problems
• Solve problems using the properties of special quadrilaterals, such as the square, rectangle, parallelogram, rhombus, and trapezoid, and describe relationships among these sets of special quadrilaterals.
• Solve problems using the properties (e.g., angles, sum of angles, number of diagonals, and vertices)of polygons with more than four sides.
• Solve problems using the properties of circles, including those involving inscribed angles, central angles, chords, radii, tangents, secants, arcs, and sectors.
•        Compute the perimeter and areas of triangles, quadrilaterals, and circles, and regions that are combinations of these figures.
• Use relationships (e.g., congruency, similarity) among two-dimensional geometric figures and among therr-dimensional figures to solver problems.
• Compute the surface area and volume of right prisms, pyramids, and cones, cylinders, and spheres, and solids that are combinations of these figures.
• Solve problems involving reflections, rotations, and translations of points, lines, or polygons in the plane.
• Execute geometric constructions using straight-edge and compass (e.g., bisect an angle, erect a perpendicular) and probe that the constructions yield the desired results.

     C) Trigonometry

•       Identify the relationships between radian measures and degree measures of angles.
•       Define  and use the six basic trigonometric relations on the right triangle and as defined on the unit circle using radian measure.
• Solve the problems involving right triangles that have angles
• Apply the law of sines and the law of cosines in the solution of problems. Recognize the graphs of six basic trigonometric functions and identify their period, amplitude, phase displacement or shift, and asymptotes.
• Apply the formulas for trigonometric functions of            and           in terms of trigonometric functions of x and y
• Prove identities using basic trigonometric identities
• Solve trigonometric equations and inequalities.
• Given a point in the rectangular coordinate system, identify the corresponding point in the polar coordinate system.
• Find the trigonometric form of complex numbers and apply De Moivre's theorem

o

      D) Functions and Their Graphs

•    Understand function notation for functions of one variable and be able to work with algebraic definition of a function (i.e., for every x there is at most one y) and be able to identify whether a graph in the plane is the graph of a function.
• To be able to form an equation to a graph and vice versa.
• Use the definition of a function as a mapping and be able to work with functions given in this way (e.g.,

          f : (x,y)

•  Find the domain and range of a function.
• Use the properties of algebraic, trigonometric, logarithmic, and exponential functions to solve problems (e.g., finding composite functions and inverse functions).
• Find the inverse of a one-to-one function in simple cases and know why one-to-one functions have inverses.
• Determine the graphical properties and sketch a graph of a linear, step, absolute value, or quadractic function (e.g., slope, intercepts, intervals of increase or decrease, axis of symmetry).

       E)  Probability and Statistics

•   Organize data into a presentation that is appropriate for solving a problem (e.g., construct a histogram and use it in the calculation of probabilities).
• Solve probability problems involving finite sample spaces by actually counting outcomes appropriately.
• Solve probability problems using counting techniques (E.g., If three cards are drawn from a standard deck of 52 cards, what is the probability that all three are aces?)
• Solve probability problems involving independent trials.(E.g., If a coin is tossed 5 times, what is the probability that heads occur at least 3 times)
• Solve problems using binomial distribution and be able to determine when the use of binomial distribution is appropriate.
• Solve problems involving joint probability.
• Find and interpret common measures of central tendency (population mean, sample mean, median, mode) and know which is the most meaningful to use in a given situation.
• Find and interpret common measures of dispersion (range, population standard deviation, sample standard deviation, population  variance, sample variance).
• Model an applied problem by using mathematical expectation of an appropriate discrete random variable (e.g., fair coins, expected winnings, expected profits).
• Solve problems using normal distribution.
• Solve basic intuitive problems using the concepts of uniform and chi-square distributions (no technical vocabulary)
• Recognize a valid test to determine whether to accept or reject a given null hypothesis H .

        F) Analytical Geometry

•  Determine the equations of lines and planes, given appropriate information.
• Make calculations in 2-space or 3-space (e.g., distance between two points, the coordinates of midpoint of line segment, distance between a point and a plane).
• Given a geometric definition of a conic section, derive the equation for the conic section (e.g., given that a parabola is the set of points that are equidistant from a given point and a given line, derive its equation).
• Determine which conic section is represented by a given equation if the axis of symmetry is parallel to one of the coordinate axes and if no restrictions are placed on location of theplane.

        G) Calculus

• Discuss informally what it means for a function to have a limit at a point.
• Calculate limits of functions or determine that the limit does not exist.
• Solve problems using properties of limits
• Use limits to show that a particular function is continuous.
• Use L'Hospital's rule, where applicable, to calculate limits of functions.
• Relate the derivative of a function to a limit or to the slope of a curve.
• Explain conditions under which a continuous function does not have a derivative.
• Differentiate algebraic expressions, trigonometric functions, and exponential and logarithmic functions using the sum, product, quotient, and chain rules.
• Use implicit differentiation.
• Make numerical approximations of derivatives and integrals.
• Use differential calculus to analyze the behavior of a function (e.g., find relative maxima and minima, concavity).
• Use differential calculus to solve problems involving related rates and rates of change.
• Approximate roots of functions (e.g., using Newton's method with derivatives).
• Use differential calculus to solve applied minima-maxima problems.
• Solve problems using the mean value theorem of differential calculus.
• Explain the significance of, and solve problems using, the fundamental theorem of calculus
• Demonstrate an intuitive understanding of the process of integration as finding areas of regions in the plane through limiting process.
• Integrate functions using standard integration techniques.
• Evaluate improper integrals..
• .Use integral calculus to calculate the area of regions in the plane and the volume of solids formed by rotating plane figures about a line.
• Determine the limits of sequences and simple infinite series.

Use standard tests to show convergence (either conditional or absolute) or divergence of series (e.g., comparison, ratio, etc).