- Foundation year
Pre-calculus: the bridge from school
Functions and their graphs, exponential and logarithmic equations, and trigonometric identities, taught so that calculus has solid ground. Marks slip on domain restrictions, sign errors and a decimal written where the course wants an exact value.
- Functions and graphs
- Logarithmic equations
- Trigonometric identities
- Calculus I
Limits and derivatives
Limits and continuity, differentiation rules up to the chain rule, implicit differentiation, related rates and optimisation. Most students manage the rules; trouble starts when a word problem must first become a function.
- Limits and continuity
- Chain rule
- Related rates and optimisation
- Calculus I and II
Integrals and integration techniques
Antiderivatives, definite integrals and the Fundamental Theorem, then the techniques (substitution, partial fractions, integration by parts) applied to area and volume. The hard part is choosing a technique and changing the limits after a substitution.
- Fundamental Theorem
- Integration by parts
- Partial fractions
- Calculus II
Series, parametric and polar work
Sequences, infinite series, power series and Taylor series, followed by parametric equations and polar coordinates. Students lose marks picking a convergence test by guesswork and skipping the endpoints of an interval of convergence.
- Convergence tests
- Taylor series
- Polar coordinates
- Calculus III
Calculus in several variables
Partial derivatives, gradients, tangent planes and double and triple integrals, with vector calculus where your syllabus includes it. Sketching a region in three dimensions and setting the limits of integration cause most lost marks.
- Partial derivatives
- Multiple integrals
- Change of coordinates
- First or second year
Linear algebra
Systems of linear equations, row reduction, matrices and determinants, vector spaces and subspaces, eigenvalues, eigenvectors and linear transformations. The arithmetic comes quickly; words such as span, basis and independence take longer to use precisely.
- Row reduction
- Vector spaces
- Eigenvalues and eigenvectors
- Second year
Differential equations for engineers and scientists
First-order methods, second-order linear equations with constant coefficients, undetermined coefficients, variation of parameters, systems solved with eigenvalues, and Laplace transforms. Recognising which type of equation sits in front of you is half the work.
- Second-order linear equations
- Laplace transforms
- Linear systems
- Engineering, science and business
Probability and statistics
Descriptive statistics, probability rules, binomial, Poisson and normal distributions, sampling distributions, confidence intervals, hypothesis tests and regression, often with software labs. Answers fall short when the test is chosen by habit or the result is left unexplained.
- Probability distributions
- Hypothesis testing
- Linear regression
- Business degrees
Mathematics for business and economics
Mathematics of finance, matrices and linear systems, linear programming, partial derivatives and constrained optimisation, with integration applied to economic quantities. Students often follow the algebra, then misread what a derivative says about cost or revenue.
- Mathematics of finance
- Linear programming
- Constrained optimisation
Your lecturer's syllabus comes first. Send it with your enquiry, and the tutor follows its chapter order, its notation and the methods your lecturer prefers.