Here are my mathematics course notes as of July 2018. You are free to use these for self-study. You may not use these for commercial purposes.
- Basic algebra, equations, formulae and identities
- Linear inequalities and intervals
- Simultaneous (linear) equations
- Rules for indices
- Rules for logarithms
Straight lines and graphs
Quadratic equations and graphs
- Quadratics: factorising, completing the square, formula
- Quadratic discriminant.
- Sketching quadratic graphs. Solving quadratic inequalities.
Trigonometric equations and identities
- Basic trig functions and graphs (sine, cosine and tangent)
- Solving trig equations with sine, cosine and tangent
- Further trig equations (including reciprocal and multiple angle functions)
- Pythagorean identities
- Transformations of trig graphs
- Properties of trig functions and further identities
- Proving identities
- \( a \sin x + b\cos x \)
Polynomial division. Factor and remainder theorem
- Functions. Domain and range.
- Composite functions
- Inverse functions
- Rational functions and graphs
- Modulus function
- Modulus equations and inequalities
- Differentiation from first principles
- Standard derivatives
- Chain rule
- Product and quotient rules
- Implicit differentiation
- Logarithmic differentiation
Sequences and series
- Linear recurrence relations
- Sigma notation
- Arithmetic series
- Geometric series
- Binomial series (part 1)
- Binomial series (part 2)
- Binomial series (part 3)
- Maclaurin series
- Partial fractions
- Vector geometry
- Cartesian vectors
- Vectors in 3D
- Scalar product. Angle between vectors.
- Vector equation of a line
- Pairs of vector lines
- Integration from first principles. Standard integrals.
- Constant of integration. Boundary conditions.
- Definite integration
- Reverse chain rule
- Integration by substitution
- Integration by parts
- Integration of partial fractions
- Integration of trig identities
Numerical solution of equations
- Interval bisection method
- Newton-Raphson method
- Fixed point iteration. Cobweb and staircase diagrams.
- Solving differential equations by separation of variables
- Growth and decay equations
- Direction fields
- Partial differentiation
- Second derivatives. Mixed derivative theorem.
- Small increments. Sensitivity analysis.
- Vector kinematics. Differentiating and integrating vectors.
- Circular motion
- Simple harmonic motion
- Shifted SHM
Polar equations and geometry