Calculus
Sequences
What a sequence is and how it is written. Arithmetic and geometric sequences, finding the common difference, the sum of terms and the basic formulae.
Limit of a sequence
The limit of a sequence is the number that the sequence approaches. The definition, convergent and divergent sequences, and methods for working out limits.
Limits
The limit of a function describes how a function behaves near a point, even where it is not defined. Neighbourhoods, accumulation points, the definition of a finite limit and examples.
Continuity of a function
What a continuous function is and how it is defined using limits. Discontinuities: removable discontinuities and discontinuities of the first and second kind.
Derivative of a function
What the derivative of a function is and what it expresses. Deriving the definition from the limit of a function, finite and infinite derivatives and basic properties of differentiation.
Introductory derivative examples
Introductory worked examples on derivatives. Formulae for the derivatives of polynomials, sums, products, quotients and composite functions, with the working shown.
Advanced derivative examples
Worked examples of harder derivatives: composite functions, exponentials and quotients. The working is explained step by step through to the result.
Derivative formulae
An overview of formulae for differentiating: basic formulae, sums, products and quotients, logarithms, exponential and trigonometric functions in one place.
Equation of a tangent
How to write the equation of the tangent to the graph of a function at a given point using the derivative. The formula, worked examples, a tangent parallel to a line, and the equation of the normal.
Curve sketching
How to investigate the behaviour of a function step by step. Domain, monotonicity, extrema, convexity and concavity, and finding points of inflection.
Integrals
Integration as the reverse process of differentiation. What an antiderivative is, integrals of elementary functions, the integral of a sum and taking out a constant.
