Module Catalogue 2025/26

Pre Requisite Comment

N/A

Co Requisite Comment

N/A

Aims

To introduce students to rigorous analysis through the study of properties of the Real Numbers and to put some of the results they have seen in MSP1612 on a firmer footing.

Module Summary
Calculus, as enlightenment thinkers such as Newton and Leibniz thought of it, was very successful at solving problems involving elementary functions. However, as mathematicians began to consider more sophisticated and complicated functions in the 18th and 19th centuries, they needed methods of ensuring that the results they needed still held true. This module introduces students to ways of thinking about the Real Numbers that allow them to prove rigorously some of the results they will have seen at school and in MSP1612 as well as laying the foundation for doing analysis and calculus with mathematical objects other than Real Numbers (e.g. Complex Numbers).

Outline Of Syllabus

Properties and constructions of the Real Numbers
Sequences and convergence
Series
Functions and limits
Continuity and its consequences
Differentiability and its consequences
Power series
The Riemann integral

Intended Knowledge Outcomes

Students will be able to reproduce, explain, and apply the precise definitions of key concepts in analysis, in particular that of convergence and continuity. They will learn the statement and proofs of the key results concerning sequences, series, continuous or differentiable functions. They will learn the definition of the Riemann integral and some examples of functions that can and cannot be integrated according to the definition.

Intended Skill Outcomes

Students will be able to compute limits of sequences and series. They will be able to write short proofs of results and identify erroneous steps in arguments.

Students will develop skills across the cognitive domain (Bloom's taxonomy, 2001 revised edition): remember, understand, apply, analyse, evaluate and create.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. 

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of programme accreditation. Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

General Notes

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