Warwick MA142 Calculus 1: Syllabus, Topics and Practice

MA142 Calculus 1 is a Year 1 module at the University of Warwick, taught by the Warwick Mathematics Institute. It is worth 10 credits and runs for 10 weeks. In the catalogue's own words, calculus "usually stands for Analysis, focused on calculations rather than proving theorems", and MA142 is the first module in that series. This guide maps each MA142 topic to free practice, so you can learn every idea by solving problems with instant feedback.

What Warwick MA142 covers

The official module catalogue gives a short, indicative outline syllabus: real numbers, limits, series, continuity and uniform continuity. The stated aim is to develop analysis in dimension one "with much greater precision and rigour" than at school, shifting the focus from problem-solving recipes to the concepts behind them.

The learning outcomes say that by the end of the module you should:

• understand the real numbers and the symbol \infty in depth,
• have a working knowledge of sequences and series, including limits and conditional and absolute convergence,
• know the properties of continuous functions.

Compared with a typical first-semester Calculus 1 course, MA142 spends less time on derivative rules and more on limits, sequences and series. Derivatives and integrals arrive in the follow-on module, MA143 Calculus 2. The catalogue does not list formal prerequisites or the term the module runs in, so check your course handbook for those.

MA142 topic map

Uniform continuity is a proof-based idea that goes beyond what our lessons cover. Use the continuity lessons to make sure the basic definition is automatic, then work through your lecture notes for the uniform version.

How MA142 fits with MA143 and MA259

See the MA143 guide for the next step.

Example problems at the MA142 level

1. A limit of a sequence. Find \lim_{n \to \infty} \dfrac{3n^2 + 1}{n^2 + 5n}. Divide top and bottom by n^2:

2. A geometric series. Since the ratio is \tfrac{1}{2} and |\tfrac{1}{2}| < 1,

3. Absolute versus conditional convergence. The alternating harmonic series \sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n} converges by the alternating series test, but \sum \dfrac{1}{n} diverges, so the convergence is conditional, not absolute.

How to do well in MA142

• Practise a little every day. Limits and series reward repetition far more than last-minute revision, and the exam is worth 85% of the module.
• Learn the standard limits by heart. Facts such as \frac{1}{n} \to 0 and r^n \to 0 for |r| < 1 appear in almost every series argument.
• Use the homework. The catalogue notes that marked work is returned and discussed in small classes, which is the best place to fix gaps early.
• Watch out for common slips. Our list of common calculus mistakes covers the ones that cost marks most often.

Tip: if a series question feels stuck, ask first whether the terms tend to zero. If they do not, the nth term test settles it straight away.