Complex numbers and the Argand diagram

Arithmetic of complex numbers

Complex roots of quadratic equations

Complex roots of cubic and quartic equations

Sum and product of roots

Manipulating expressions in the roots

Forming quadratic equations with new roots

Interval bisection and linear interpolation

The Newton-Raphson process

The parabola

The rectangular hyperbola

Tangents and normals to the parabola and rectangular hyperbola

Matrix arithmetic

Determinants and inverses of 2x2 matrices

Linear transformations as matrices

Matrix representations of standard transformations

Combined and inverse transformations

Summation of finite series

Proof by mathematical induction

Algebraic and modulus inequalities

The method of differences

Euler's relation and polar form

De Moivre's theorem and trigonometric identities

Roots of complex numbers

Loci and regions in the Argand diagram

Transformations from the z-plane to the w-plane

Separable differential equations and families of solutions

First order linear differential equations

Differential equations reducible by substitution

Constant coefficient second order equations

Second order equations reducible by substitution

Maclaurin series

Taylor series

Series solutions of differential equations

Polar coordinates and curve sketching

Area in polar coordinates

Hyperbolic functions and identities

Inverse hyperbolic functions

The ellipse and hyperbola

Focus-directrix properties and eccentricity

Tangents, normals and loci

Differentiating hyperbolic functions

Differentiating inverse trigonometric and hyperbolic functions

Integrating hyperbolic and inverse functions

Hyperbolic and trigonometric substitutions

Reduction formulae

Arc length and surface area of revolution

The vector product

Lines and planes in three dimensions

Three-dimensional transformations and 3x3 matrices

Eigenvalues and eigenvectors

Diagonalisation of symmetric matrices

Motion in a vertical plane and projectiles

Kinematics using calculus

Centre of mass of discrete masses

Centre of mass of laminas and composite figures

Equilibrium of a lamina

Work, energy and power

Conservation of mechanical energy

Momentum and impulse in vector form

Direct impact and Newton's law of restitution

Successive impacts

Moments and equilibrium of rigid bodies

Variable acceleration in a straight line

Hooke's law

Elastic potential energy

Variable forces

Simple harmonic motion

Oscillations of elastic strings and springs

Angular speed and radial acceleration

Horizontal circular motion

Motion in a vertical circle

Centres of mass of uniform bodies by integration

Equilibrium of rigid bodies

The binomial distribution

The Poisson distribution

Poisson approximation to the binomial

Probability density and cumulative distribution functions

Mean, variance, mode, median and quartiles

The continuous uniform distribution

Normal approximation to the binomial and Poisson

Populations, censuses and samples

Hypothesis testing concepts

Hypothesis tests for binomial and Poisson

Linear combinations of Normal random variables

Methods of sampling

Estimators, bias and the sampling distribution of the mean

Confidence intervals for a Normal mean

Hypothesis tests for a mean

Hypothesis tests for the difference between two means

Chi-squared goodness of fit tests

Spearman's rank correlation coefficient

Testing for zero correlation