REAL ANALYSIS 2  2025/6
Module code: MAT2004
Module Overview
Analysis is the branch of mathematics that rigorously studies functions, continuity, differentiability and integration. This module builds on Level 4 Real Analysis 1 (MAT1032) by studying these concepts in a more formal way and hence provides a deeper understanding of these concepts.
Module provider
Mathematics & Physics
Module Leader
GRANT James (Maths & Phys)
Number of Credits: 15
ECTS Credits: 7.5
Framework: FHEQ Level 5
Module cap (Maximum number of students): N/A
Overall student workload
Independent Learning Hours: 59
Lecture Hours: 33
Tutorial Hours: 10
Guided Learning: 15
Captured Content: 33
Module Availability
Semester 1
Prerequisites / Corequisites
N/A
Module content
Indicative content includes:
Limits and continuity of the sums, products and composition of realvalued functions;
Intermediate Value Theorem, Extreme Value Theorem;
Derivatives of the sums, products, quotients, composition and inverse of realvalued functions;
Rolle’s Theorem, Mean Value Theorem, L’Hôpital’s Rule;
Higher derivatives, Taylor’s Theorem, Contraction Mapping Theorem;
Upper and lower sums of integrals, the Riemann integral;
Indefinite integration, Fundamental Theorem of Caculus, Taylor series with integral remainder.
Assessment pattern
Assessment type  Unit of assessment  Weighting 

Schooltimetabled exam/test  Insemester test (50 minutes)  20 
Examination  Exam (2 hours)  80 
Alternative Assessment
N/A
Assessment Strategy
The assessment strategy is designed to provide students with the opportunity to demonstrate:¿
Understanding, interpretation and manipulation of mathematical statements.
Subject knowledge through implicit recall of key definitions, theorems and their proofs.
Analytical ability through logical proofs or counterexamples to unseen problems in the test and exam.
Thus, the summative assessment for this module consists of:
One insemester test taken during the semester, worth 20% of the module mark, corresponds to Learning Outcome 1, 2, 3.
A synoptic examination (2 hours), worth 80% of the module mark, corresponds to Learning Outcomes 1, 2, 3.
Formative assessment
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.
Feedback
Individual written feedback is provided to students for formative unassessed courseworks. The feedback is timed such that feedback from the first coursework will assist students with preparation for the insemester test. The feedback from both courseworks and the insemester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback during lectures and tutorials.
Module aims
 Extend students' understanding of realvalued functions by studying their properties in a formal way.
 Enable students to determine the properties of a given function from first principles.
 Facilitate students' understanding of analysis through frequently encountered examples, counterexamples, problems and applications.
Learning outcomes
Attributes Developed  
001  Students will understand realvalued functions and be able to prove limits, continuity, differentiability and integrability using the formal definitions and basic properties.  KC 
002  Students will be able to quote, prove and apply main theorems, including (but not limited to) the Intermediate Value Theorem, Extreme Value Theorem, Rolle's Theorem, Mean Value Theorem, L'Hôpital's Rule, Taylor's Theorem and Fundamental Theorem of Calculus.  KC 
003  Students will be able to give logical proofs or counterexamples to unseen statements relating to limits, continuity, differentiability and integrability.  KCT 
Attributes Developed
C  Cognitive/analytical
K  Subject knowledge
T  Transferable skills
P  Professional/Practical skills
Methods of Teaching / Learning
The learning and teaching strategy is designed to:
Give a detailed introduction to realvalued functions, limits, continuity, derivatives and integrals and ensure experience in the methods used to interpret, understand and solve problems in analysis.
The learning and teaching methods include:
Three onehour lectures per week for eleven weeks, with typeset notes to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.
Ten tutorials for guided discussion of solutions to problem sheets (provided to students in advance for completion to reinforce their understanding and guide their learning).
Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.
Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students the opportunity to review parts of lectures that they may not fully have understood and should not be seen as an alternative to attending lectures.
Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include inclass tests where one or more of these are an assessment on the module. Inclass tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.
Reading list
https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MAT2004
Other information
The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MAT2004 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies and facilitates collaborative learning and information sharing.

Employability: The module MAT2004 equips students with skills which significantly enhance their employability. Students gain mathematical proficiency, which hones critical thinking and problemsolving abilities. Students learn to evaluate complex problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

Global and Cultural Capabilities: Students enrolled in MAT2004 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problemsolving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MAT2004 is a module which demands a rigorous approach to Real Analysis, to which students will learn to adapt. They will gain skills in analysing problems and lateral thinking. Students will complete assessments which challenge them and build resilience.
Programmes this module appears in
Programme  Semester  Classification  Qualifying conditions 

Mathematics with Statistics BSc (Hons)  1  Optional  A weighted aggregate mark of 40% is required to pass the module 
Mathematics BSc (Hons)  1  Compulsory  A weighted aggregate mark of 40% is required to pass the module 
Mathematics MMath  1  Compulsory  A weighted aggregate mark of 40% is required to pass the module 
Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.