MA2RAT: Real Analysis II

Module code: MA2RAT

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: Level 2 (Intermediate)

When you'll be taught: Semester 2

Module convenor: Dr Nikos Katzourakis, email: n.katzourakis@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST ( TAKE MA1FM AND TAKE MA1RA1 AND TAKE MA1CA ) OR ( TAKE MA0FMNU AND TAKE MA1DE1NU AND TAKE MA1RA1NU ) (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s):

Module(s) excluded:

Placement information: NA

Academic year: 2024/5

Available to visiting students: Yes

Talis reading list: Yes

Last updated: 21 May 2024

Overview

Module aims and purpose

The aim of thismodule is to continue the investigations into rigorous mathematical analysis, which was initiated in the pre-requisite module on real analysis. 

Module learning outcomes

By the end of the module, it is expected that students will be able to:

1. Understand and use the ideas of monotonicity, limits, continuity, integration and differentiation in a rigorous way
2. Manipulate infinite series and apply them in problems involving differentiation, integration, sequences of functions and series
3. Understand that there are situations in which the order of limiting operations may not be reversed, and justify some simple cases where the inversion is legitimate. 
4. Understand and be able to construct basic rigorous arguments involving the main notions of convergence, topology, continuity and differentiability offunctions in the Euclidean space. 

Module content

Analysis in one variable: real valued functions of a real variable, limits, continuity, monotonicity, differentiation, Taylor expansions, Riemann integration, improper integrals, sequences of functions, pointwise and uniform convergence, series of functions, power series, smooth and analytic functions.  

Analysis in several variables: norms and convergence of sequences in the Euclidean space, elementary notions of topology, real and vector functions of several variables, continuity and differentiation, Riemann integration in several variables. 

Structure

Teaching and learning methods

Lectures supported by exercise sheets and tutorials.

Study hours

At least 54 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 40% to pass this module.

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Weekly exercise sheets and tutorials.

Reassessment

Additional costs