Course outline

Aim

• To build confidence and understanding of mathematical ideas.
• To enable the student to interpret and solve mathematical problems.
• To improve examination technique and to make the student aware of common examination errors.
• To make the student fully aware of the content and emphasis of the specification.
• To strengthen revision methods.

Key Topics Covered

1. Algebra and functions

• Laws of indices for all rational exponents.
• Use and manipulation of surds.
• Quadratic functions and their graphs.
• The discriminant of a quadratic function.
• Completing the square.
• Solution of quadratic equations.
• Simultaneous equations: analytical solution by substitution.
• Solution of linear and quadratic inequalities.
• Algebraic manipulation of polynomials, including expanding brackets, collecting like terms and factorisation.
• Graphs of functions; sketching curves defined by simple equations.
• Geometrical interpretation of algebraic solution of equations.
• Use of intersection points of graphs of functions to solve equations.
• Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y =af(x),y=f(x)±a,y=f(x±a),y=f(ax).

2. Coordinate geometry in the (x, y) plane

• Equation of a straight line, including the forms y - y₁ = m(x-x₁) and ax + by + c = 0.
• Conditions for two straight lines to be parallel or perpendicular to each other.

3. Sequence and series

• Arithmetic series, including the formula for the sum of the first n natural numbers.
• Sequences, including those given by a formula for the n^th term and those generated by a simple relation of the form x_n+1=f(x_n).

4. Differentiation

• Differentiation of xⁿ, and related sums and differences.
• The derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point; the gradient of the tangent as a limit; interpretation as a rate of change; second order derivatives.
• Applications of differentiation to gradients, tangents and normals.

5. Integration

• Integration of xⁿ.
• Indefinite integration as the reverse of differentiation.
• Finding y=f(x) given f’(x) and a point on the line.