Edinburgh MATH08081 Introduction to Mathematical Analysis: Guide

MATH08081 Introduction to Mathematical Analysis is the Year 1 analysis course in the School of Mathematics at the University of Edinburgh, taught in Semester 2. It is worth 20 SCQF credits (10 ECTS). The course catalogue calls analysis "the theory underpinning calculus": you prove why limits, continuity and derivatives behave the way you learned at school. This guide maps each MATH08081 topic to free practice so you can keep your calculus skills sharp while the proofs take centre stage.

What MATH08081 covers

The Edinburgh course catalogue (DRPS) gives a representative outline: real numbers, sequences, infinite series, limits, continuous functions and differentiation. It says there is a strong focus on rigorous mathematics, "quite different to how mathematics is typically taught at school", building on the skills from Introduction to Mathematics at University (MATH08078) in Semester 1. It also says students will get plenty of chances to practise and reinforce their calculus skills.

The learning outcomes add detail. By the end you should understand the real numbers and the completeness axiom, limits in epsilon-N language, infinite sequences and series and their convergence, limits and continuity in epsilon-delta language, and differentiation. You also work with rigorous definitions of the exponential, trigonometric and logarithmic functions, and derive existence and uniqueness results for very simple linear ODEs with constant coefficients.

Compared with a North American Calculus 1, MATH08081 covers limits, continuity and derivatives plus sequences and series, but not integration, which comes in MATH08082. The emphasis is on proof.

MATH08081 topic map

Where MATH08081 fits

Engineering students take Engineering Mathematics 1b (MATH08075) instead, which is more computational and cannot be combined with MATH08081.

Example problems at the MATH08081 level

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

In the course you would also prove it: for any \varepsilon > 0, find N so that the distance to 3 is below \varepsilon for all n > N.

2. A convergence test. Does \sum_{n=1}^{\infty} \frac{n}{2^n} converge? By the ratio test,

so the series converges.

3. A derivative from the definition. For f(x) = x^2,

How to do well in MATH08081

• Keep computing while you learn to prove. Proofs are much easier when the calculation behind them is automatic. Short daily sessions on limits and series help.
• Learn the definitions word for word. Most exam questions start from the precise definition of a limit, continuity or the derivative.
• Work every tutorial sheet. The catalogue stresses engaging with all the learning opportunities, not just lectures, and coursework is 20% of the mark.
• Watch for common slips. Our list of common calculus mistakes covers the algebra errors that cost marks.

Tip: the squeeze theorem is one of the most useful tools in a first analysis course. Practise spotting when a sequence can be trapped between two simpler ones.