Understand the meaning of |x|, sketch the graph of y = |ax + b|, and use relations such as |a| = |b| ⇔ a2 = b2 and |x - a| < b ⇔ a - b < x < a + b when solving equations and inequalities (graphs of y = |f(x)| and y = f(|x|) for non-linear functions f are not included).
Divide a polynomial of degree not exceeding 4 by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero).
Use the factor theorem and the remainder theorem.
Logarithmic and Exponential Functions
Understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base).
Use logarithms to solve equations and inequalities in which the unknown appears in indices.
Use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept.
Understand the relationship between ex and ln x, and their graphs (include the graph of y = ekx for both positive and negative values of k).
Trigonometry
Understand the relationship of the sine, cosine, and tangent functions to cosecant, secant, and cotangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude.
Use trigonometric identities to simplify expressions and solve equations:
sin2θ + cos2θ = 1
sec2θ = 1 + tan2θ
cosec2θ = 1 + cot2θ
expansions of sin(A ± B), cos(A ± B), and tan(A ± B)
formulae for sin2A, cos2A, and tan2A
expression of asinθ + bcosθ in the forms Rsin(θ ± \(\alpha\)) and Rcos(θ ± \(\alpha\))
Differentiation
Use the derivatives of ex, ln x, sin x, cos x, tan x, together with constant multiples, sums, differences, and composites.
Differentiate products and quotients.
Use the first derivative to solve problems involving tangents and normals.
Integration
Integrate functions of the form eax + b, \(\frac{1}{ax + b}\), sin(ax + b), cos(ax + b), and sec2(ax + b) (knowledge of the general method of integration by substitution is not required).
Use the trapezium rule to estimate the value of a definite integral (sketch graphs in simple cases to determine whether the trapezium rule gives an under-estimate or an over-estimate).
Numerical Solution of Equations
Find the root of an equation by means of graphs and/or searching for a sign change (e.g., finding a pair of consecutive integers between which a root lies).
Use the notation for a sequence of approximations which converges to a root of an equation.
Use a given iteration form xn+1 = F(xn), or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.
