Course Content

CSEC Additional Mathematics SBA

Algebra and Functions: Algebra
 Perform operations of addition, subtraction, multiplication and division of polynomial and rational expressions
 Factorize polynomial expressions, of degree less than or equal to 4, leading to real linear factors
 Apply the Remainder Theorem
 Use the Factor Theorem to find factors and to evaluate unknown coefficients

Algebra and Functions: Quadratics
 Express the quadratic function ” ax^2 + bx + c = 0″ in the form “a (x + h) j + k = 0”, where h and k are constants to be determined
 Determine maximum or minimum values and range of quadratic functions by completion of the square
 Sketch the graph of the quadratic function, including maximum or minimum points
 Determine the nature of the roots of a quadratic equation
 Solve equations in x reducible to a quadratic equation, for example, x^4 – 6x^2 + 8 = 0 and x – 2√x + 1 = 0
 Use the relationship between the sums and products of the roots and the coefficients of ax^2 + bx + c = 0
 Solve two simultaneous equations in 2 unknowns in which one equation is linear and the other equation is quadratic

Algebra and Functions: Inequalities

Algebra and Functions: Functions
 Use terms related to functions
 Determine the range of a function given its domain
 Determine whether a given function is manytoone or onetoone
 Determine the inverse of a given function, (if it exists)
 Plot and sketch functions and their inverses, (if they exist)
 State the geometrical relationship between the function y = f(x) and its inverse f^1( x )
 Find the composition of two functions
 Recognize that, if g is the inverse of f, then f [g(x)] = x, for all x, in the domain of g

Algebra and Functions: Surds, Indices, and Logarithms
 Perform operations involving surds
 Use the laws of indices to solve exponential equations with one unknown
 Use the fact that loga b = c <> a^c = b where a is any positive whole number
 Solve logarithmic equations
 Use logarithms to solve equations of the form a^x = b
 Apply logarithms to problems Linear Reduction, involving the transformation of a given relationship to linear form

Algebra and Functions: Sequences and Series
 Define a sequence of terms {an} where n is a positive integer
 Write a specific term from the formula for the nth term of a sequence
 Use the summation (Σ) notation
 Define a series, as the sum of the terms of a sequence
 Identify arithmetic and geometric series
 Obtain expressions for the general terms and sums of finite arithmetic and finite and infinite geometric series
 Show that all arithmetic series (except for zero common difference) are divergent, and that geometric series are convergent only if 1 < r <1, where r is the common ratio
 Calculate the sum of arithmetic series to a given number of term
 Calculate the sum of geometric series to a given number of term
 Find the sum of a convergent geometric series

Coordinate Geometry, Vectors and Trigonometry: Coordinate Geometry
 Find the equation of a straight line
 Determine whether lines are parallel or mutually perpendicular using the gradients
 Find the point of intersection of two lines
 Write the equation of a circle
 Find the centre and radius of a given circle
 Find equations of tangents and normals at given points on circles
 Find the points of intersection of a curve with a straight line

Coordinate Geometry, Vectors and Trigonometry: Vectors
 Define equal vectors
 Add and subtract vectors
 Multiply a vector by a scalar quantity
 Derive and use unit vectors
 Find displacement vectors
 Find the magnitude and direction of a vector
 Define the scalar product of two vectors: (i) in terms of their components (ii) in terms of their magnitudes and the angle between them
 Find the angle between two given vectors
 Apply properties of parallel and perpendicular vectors

Coordinate Geometry, Vectors and Trigonometry: Trigonometry
 Define the radian
 Convert degrees to radians and radians to degrees
 Use the formulae for Arc Length “l = rq” and Sector Area “A= (½)(r^2)(q)”
 Evaluate sine, cosine and tangent for angles of any size given either in degrees or radians
 Graph the functions sin kx, cos kx, tan kx, where k is 1 or 2 and “0 ≤ x ≤ 2π”
 Derive the identity cos^2(0) + sin^2(0) = 1
 Use the formulae for sin(A±B), cos(A±B) and tan(A±B)
 Derive the multiple angle identities for sin 2x, cos 2x, tan 2x
 Use Lessons 7, 8 and 9 above to prove simple identities
 Find solutions of simple trigonometric equations for a given range, including those involving the use of cos^2(q) + sin^2(q) = 1 .

Introductory Calculus: Differentiation
 Use the concept of the derivative at a point x = c as the gradient of the tangent to the graph at x = c
 Define the derivative at a point as a limit
 Use the f'(x) and dy/dx notation for the dx first derivative of f(x)
 Use d/dx(x^n) = n(x^n1) where n is any real number
 Use “d/dx(sin x) = cos x” and “d/dx(cos x) = sin x”
 Use simple rules of derivatives to find derivatives of sums and multiples of functions
 Use Lessons 4, 5 and 6 above to calculate derivatives of polynomials and trigonometric functions
 Apply the chain rule in the differentiation of composite functions
 Differentiate products and quotients of simple polynomials and trigonometric functions
 Use the concept of the derivative as a rate of change
 Use the concept of stationary points
 Determine the nature of stationary points
 Locate stationary points, maxima and minima, by considering sign changes of the derivative
 Calculate the second derivative, f “(x)
 Interpret the significance of the sign of the second derivative
 Use the sign of the second derivative to determine the nature of stationary points
 Obtain equations of tangents and normals to curves

Introductory Calculus: Integration
 Recognize integration as the reverse process of differentiation
 Show that the indefinite integral represents a family of functions which differ by constants
 Use simple rules of integration
 Integrate functions of the form “(ax ± b)^n” where a, b, n are real and n ≠ 1
 Find indefinite integrals using formulae and integration theorems
 Integrate simple trigonometric functions
 Compute definite integrals
 Formulate the equation of a curve given its gradient function and a point on the curve
 Apply integration to find the area of the region in the first quadrant bounded by a curve and the lines parallel to the yaxis
 Apply integration to find volumes of revolution polynomials up to and including degree 2

Basic Mathematical Applications: Data Representation and Analysis
 Distinguish between types of data
 Represent numerical data diagrammatically
 Outline the relative advantages and disadvantages of stemandleaf diagrams and boxandwhisker plots in data analyses
 Interpret stemandleaf diagrams and boxandwhiskers plots
 Determine quartiles and percentiles from Percentiles, raw data, grouped data, stemandleaf diagrams, boxandwhisker plots
 Calculate measures of central tendency and dispersion
 Explain how the standard deviation measures the spread of a set of data

Basic Mathematical Applications: Probability Theory
 Distinguish among the terms experiment, outcome, sample space and event
 Calculate the probability of event A, P(A), as the number of outcomes of A divided by the total number of possible outcomes, when all outcomes are equally likely and the sample space is finite
 Use the basic law of probability: the sum of probabilities of all the outcomes in a sample space is equal to one
 Use the basic law of probability: 0 £ P(A) £ 1 for any event A
 Use the basic law of probability: P(A’) = 1 – P(A), where P(A’) is the probability that event A does not occur
 Use P (A È B) = P(A) + P(B) – P(A Ç B) to calculate probabilities
 Identify mutually exclusive events A and B such that P(A Ç B) = 0
 Calculate the conditional probability P(A  B) where P(A  B) = P(A∩B)/P(B)
 Identify independent events
 Use the property P(A Ç B) = P(A) x P(B) or P(A  B) = P(A) where A and B are independent events
 Construct and use possibility space diagrams, tree diagrams and Venn diagrams to solve problems involving probability

Basic Mathematical Applications: Kinematics of Motion along a straight line