2024/25 Undergraduate Module Catalogue

MATH1110 Real Analysis

20 creditsClass Size: 275

Module manager: TBC

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2024/25

Pre-requisite qualifications

Grade B in A-level Mathematics or equivalent.

This module is not approved as a discovery module

Module summary

Calculus is arguably the most significant and useful mathematical idea ever invented, with applications throughout the natural sciences and beyond. This module develops the theory of differential and integral calculus of real functions in a precise and mathematically rigorous way. Particular emphasis will be put on sequential notions.

Objectives

This module will introduce students to a rigorous understanding of sequences, series and calculus.

Learning outcomes
On successful completion of the module students will have demonstrated the following learning outcomes relevant to the subject:
1. Read and write statements in formal mathematics and be able to understand how the order of quantifiers affects the meaning of a statement.
2. Explain the meaning and differences between maximum, minimum, supremum, and infimum.
3. Know the formal definition of a limit and be able to prove that a given sequence or series is convergent or divergent.
4. Apply the definitions of continuity, sequential continuity, and differentiability, and understand
how those definitions interact with each other.
5. Use basic propositions about continuity and differentiability to prove more results about those concepts, including showing that a specific function is continuous or differentiable.
6. Recall and apply important theorems of Real Analysis, including the Intermediate Value Theorem and the Mean Value Theorem.
7. Apply definitions and results about Power Series, including finding the Taylor expansion

Skills Learning Outcomes
SLO1. Communicate through written work technical information and reasoning.
SLO2. Apply analytical thinking and technical knowledge to solve problems.
SLO3. Write in a clear, concise, and focused way.
SLO4. Manage workloads, deadlines, and workplace pressure through prioritisation and productivity skills.

Syllabus

The following topics will be covered:
1. Symbolic logic, quantifiers, and sets of numbers,
2. The supremum, infimum, maximum and minimum of a set of real numbers,
3. Sequences and limits,
4. Algebra of limits,
5. Subsequences and the Bolzano-Weierstrass Theorem,
6. Cauchy sequences,
7. Series, their convergence, non-convergence, and absolute convergence,
8. Limits and continuity of functions,
9. Differentiation,
10. Major theorems of Real Analysis, including the Intermediate Value Theorem, and the Mean Value Theorem,
11. Taylor’s Theorem and power series.

Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar:
12. Riemann integration.

Methods of Assessment

We are currently refreshing our modules to make sure students have the best possible experience. Full assessment details for this module are not available before the start of the academic year, at which time details of the assessment(s) will be provided.

Assessment for this module will consist of;

2 x Portfolio of assessed questions
2 x In-person open book exam

Teaching methods

Delivery type	Number	Length hours	Student hours
Lectures	44	1.00	44.00
Seminar	10	1.00	10.00
Independent online learning hours			48.00
Private study hours			98.00
Total Contact hours			54.00
Total hours (100hr per 10 credits)			200.00

Opportunities for Formative Feedback

Regular example sheets.

Reading list

The reading list is available from the Library website

Last updated: 30/04/2024

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