Arithmetic, Algebra and Simple Trigonometry:
· Revision of number systems and arithmetic operation
· Basic algebra, power laws . Algebraic manipulation, simplification of rational expressions, the Remainder Theorem, algebraic division; Partial fractions
· Solution of linear, quadratic equations and simultaneous linear equations
· Revision of basic trigonometry, trigonometric identities. Double angle formulae and formulae for sin (A+B) etc.
· Functions, domain, range, graph sketching, incl. straight line, circle
· Composition of functions. Piecewise defined functions. Odd, even and periodic functions.
· Inverse functions
· The exponential and logarithm functions
· The trigonometric functions and inverse trigonometric functions. Solution of trig equations
· Hyperbolic functions
Sequences and Series:
· Binomial expansions and factorial notation
· Arithmetic sequences and series. Geometric sequences and series
· Definition and use of j and complex conjugate
· Arithmetic operations with complex numbers
· Fundamental Theorem of Algebra
· Modulus and argument of a complex number and their properties. Polar and exponential representations of a complex number . Relationship between trigonometric and hyperbolic functions
· De Moivre’s Theorem
· Definition of a derivative
· Techniques of differentiation such as the product rule, the quotient rule, the chain rule
· Differentiation of a range of functions e.g trigonometric, exponential and logarithmic
· Implicit, parametric and logarithmic differentiation
· Higher derivatives
· Applications of differentiation to equations of the tangent, local maxima, minima and points of inflection, rates of change and related rates of change
· Maclaurin and Taylor series,
· L’Hopitals’ rule
· Newton Raphson method
· Indefinite integration as the reverse of differentiation
· Definite integration interpreted as the area under a curve
· Techniques of integration incl. substitution, integration by parts & using partial fractions
· Applications of integration to the area between curves.
· Numerical integration using the Trapezium rule
· Application of integration to curve lengths, surfaces and volumes of revolution, first moments and centroids, second moments and radii of gyration.
Probability and statistics:
· Descriptive statistics: numerical and graphical summaries.
- Basic Probability: elementary laws, random variables, mean and variance.