Study notes for university mathematics exams
Short, exam-oriented summaries of first- and second-year material. Written while tutoring undergraduates; kept online because students kept asking for the checklists.
How to use these notes. They are not a textbook. Each section lists what examiners actually test, the results you must be able to state from memory, and the mistakes that cost the most marks. Work through past papers alongside them.
Calculus I & II — what you must know cold
- Limits. The ε–δ definition, squeeze theorem, and the standard limits
sin x / x → 1,(1 + 1/n)^n → e. Be able to prove a limit from the definition for a linear or quadratic function. - Differentiation. Product, quotient and chain rules without hesitation; implicit differentiation; derivatives of inverse trig functions. Mean Value Theorem — statement and hypotheses.
- Integration. Substitution, integration by parts (choose
uby LIATE), partial fractions, trigonometric substitution. Know when an improper integral converges by comparison. - Series. Geometric, p-series, ratio and root tests, alternating series test. Taylor series of
e^x,sin x,cos x,ln(1+x),1/(1−x)with radius of convergence. - Multivariable. Partial derivatives, gradient, Lagrange multipliers, double integrals in polar coordinates, change of variables with the Jacobian.
| Typical question | Where marks are lost |
|---|---|
| Show that a function is continuous but not differentiable at a point | Checking only one side of the limit |
| Determine convergence of a series | Using the ratio test when the limit equals 1 and stopping there |
| Find extrema subject to a constraint | Forgetting to check boundary points and the case ∇g = 0 |
| Evaluate an improper integral | Splitting at the wrong point or ignoring a singularity inside the interval |
Linear algebra — core results
- Systems and matrices. Row reduction to RREF, rank, the rank–nullity theorem:
rank(A) + dim ker(A) = n. Invertibility criteria (nonzero determinant, full rank, trivial kernel) and why they are equivalent. - Vector spaces. Subspace test, linear independence, basis, dimension, coordinates. Change-of-basis matrices — draw the diagram before computing.
- Eigenvalues. Characteristic polynomial, algebraic vs geometric multiplicity, diagonalisability. Cayley–Hamilton. For a symmetric real matrix: real eigenvalues and an orthonormal eigenbasis.
- Inner products. Gram–Schmidt, orthogonal projection, least squares as a projection problem, Cauchy–Schwarz.
Proof questions. Examiners favour short proofs that follow directly from definitions: "if
A is invertible then its columns are linearly independent", "eigenvectors for distinct eigenvalues are independent", "the kernel is a subspace". Practise writing each in under ten lines.
Exam strategy
- Read every question first. Mark the ones you can do in your sleep and do them first — they pay the same as the hard ones.
- State the theorem you are using by name and check its hypotheses in writing. A correct answer with an unjustified step often gets half marks.
- If a computation explodes, stop. You have probably made an arithmetic slip; re-check the setup instead of pushing through.
- Leave a two-line summary at the end of long problems: what was asked, what you found. It helps the marker find your answer.
- Units and domains: an integral over an interval where the function is undefined is worth zero marks regardless of the algebra.
A four-week revision plan
| Week | Focus | Output |
|---|---|---|
| 1 | Definitions and theorem statements for the whole course | One-page summary sheet per topic |
| 2 | Worked examples from lectures, redone without notes | List of techniques you could not reproduce |
| 3 | Past papers, untimed, with full solutions written out | Error log grouped by type |
| 4 | Past papers under exam conditions | Two timed papers per week, marked honestly |
The error log matters more than the number of papers. Most students repeat the same three or four mistakes; finding yours early is worth more than another practice set.