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

• Simplification of rational expressions including factorising and cancelling, and algebraic division.
• Definition of a function; domain and range of functions; composite functions; inverse functions and their graphs.
• The modulus function.
• Combinations of the transformations y = f(x) as represented by y = af(x), y = f(x) ± a, y = f(x ± a) and y = f(ax).

2. Trigonometry

• Knowledge of secant, cosecant and cotangent and of arcsin, arccos and arctan. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains.
• Knowledge and use of sec² θ ≡ 1 + tan² θ and cosec²θ ≡ 1 + cot²θ.
• Knowledge and use of double angle formulae; use of formulae for sin(A±B), cos(A±B) and tan(A±B); use of expressions for acosθ + bsinθ in the equivalent forms of rcos(θ ± α) or rsin(θ ± α).

3. Exponentials and logarithms

• The function e^x and its graph.
• The function 1n x and its graph; 1n x as the inverse function of e^x

4. Differentiation

• Differentiation of e^x, 1n x, sin x, cos x, tan x and their sums and differences
• Differentiation using the product rule, the quotient rule and the chain rule
• The use of

5. Numerical Methods

• Location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous
• Approximate solution of equations using simple iterative methods, including recurrence relations of the form x_n+1 = f(x_n)