Manchester MATH11112 Real Analysis: Syllabus, Topics and Practice

MATH11112 Real Analysis is a Level 1 unit at the University of Manchester, taught in Semester 2 and worth 10 credits. It takes the calculus you learned at school, from series to derivatives to integrals, and puts it on rigorous foundations. The unit page says it provides "a theoretical foundation for basic calculus techniques, already familiar to the students". This guide maps each topic to free practice so your computation stays fast while you learn the proofs.

What MATH11112 covers

According to the official unit page, MATH11112 applies the fundamental properties of real numbers, sequences and limits introduced in Mathematical Foundations & Analysis (MATH11121) to study series, including power series. This allows a rigorous treatment of the exponential and trigonometric functions. The unit then studies differentiation and integration in depth and derives results such as the Fundamental Theorem of Calculus.

The learning outcomes say that by the end you should be able to:

• Establish convergence or divergence of series and find the radius of convergence of a power series.
• State the definition of differentiability, prove simple functions are differentiable and prove the rules of differentiation.
• Prove the Chain Rule, Rolle's Theorem, the Mean Value Theorem and Cauchy's Mean Value Theorem, with applications such as L'Hôpital's rule.
• Apply the Inverse Rule for monotonic differentiable functions, for example to the logarithm and powers.
• Calculate Taylor polynomials and series and apply Taylor's Theorem with an error term.
• Work with Riemann sums and upper and lower integrals, define the Riemann integral and prove the Fundamental Theorem of Calculus.

MATH11112 topic map

Where MATH11112 fits in Year 1

Example problems at the MATH11112 level

1. Radius of convergence. For \sum_{n=0}^{\infty} \frac{x^n}{2^n}, the ratio test gives

so the radius of convergence is R = 2.

2. L'Hôpital's rule. Both top and bottom tend to 0, so

3. Taylor's theorem. The Maclaurin polynomial of degree 3 for \sin x is x - \frac{x^3}{6}, and the error is bounded by \frac{|x|^5}{120} because the fifth derivative of \sin x is at most 1 in size and the degree 4 term is zero.

How to do well in MATH11112

• Learn each definition precisely. Differentiability, the Riemann integral and convergence are all tested from their exact definitions.
• Keep calculations automatic. Short daily practice on series tests and derivatives frees your attention for proofs.
• Use the supervisions. Work handed in at alternate-week supervisions is marked and discussed, which is the quickest way to fix gaps.
• Watch the common slips. Our list of common calculus mistakes covers the algebra errors that cost marks.

Tip: the Mean Value Theorem is behind L'Hôpital's rule, Taylor's theorem with remainder and the Fundamental Theorem of Calculus. If you understand its proof, much of the unit falls into place.