# REAL ANALYSIS 2 - 2023/4

Module code: MAT2004

## Module Overview

This module builds on the Year 1 module Real Analysis 1 and focuses on continuity, differentiability and integrability of real functions of one variable.

### Module provider

Mathematics & Physics

GRANT James (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 73

Lecture Hours: 33

Tutorial Hours: 11

Captured Content: 33

Semester 1

## Prerequisites / Co-requisites

MAT1032 Real Analysis 1

## Module content

This module contains the following topics:

• Limits of functions, continuity (ε-δ definition). Sums, products, compositions. Intermediate value theorem and extreme value theorem.

• Differentiable functions (sums, products, quotients). Differentiability implies continuity. Chain rule, inverse functions. Rolle's theorem, mean value theorem, l'Hôpital's rule. Higher derivatives. Taylor 's theorem. Contraction mapping theorem.

• Theory of integration: upper and lower sums and integrals, the Riemann integral. Conditions for integrability (e.g., continuity implies integrability). Indefinite integration, and the fundamental theorem of calculus. Taylor series with integral remainder.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 20
Examination Exam (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of and ability to interpret and manipulate mathematical statements.

• Subject knowledge through the recall of key definitions, theorems and their proofs.

• Analytical ability through the solution of unseen problems in the test and exam.

Thus, the summative assessment for this module consists of:

• One final examination worth 80% of the module mark.

• One In-semester test; worth 20% module mark.

Formative assessment and feedback

Students receive written feedback via marked coursework assignments over an 11 week period.  In addition, verbal feedback is provided by lecturer/class tutor at tutorial lectures.

## Module aims

• The aim of this module is to extend the introduction to real analysis by studying continuity, differentiability and integration of functions of a real variable in a more formal way and hence provide a deeper understanding of those concepts. Several applications will be presented alongside the theory.

## Learning outcomes

 Attributes Developed 1 Prove continuity, differentiability and integrability of function by using the formal definitions and basic properties. KC 2 Quote, prove and apply main theorems in Real Analysis (e.g., Intermediate, Extreme and Mean Value Theorems, Rolle's Theorem, l'Hôpital's rule, Taylor’s Theorem, Fundamental Theorem of Calculus, etc.). KC 3 Argue logically to justify proofs or give examples or counterexamples of properties of continuity, convergence, 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 provide:

• A detailed introduction to continuity, differentiability and integrability of real-valued functions.

• Experience (through demonstration) of the methods used to interpret, understand and solve problems in analysis

The learning and teaching methods include:

3 x 1 hour lectures per week x 11 weeks, with projector-displayed written notes to supplement the module handbook and Q + A opportunities for students.

1 x 1 hour interactive problem solving session/tutorial lecture per week x 11 weeks.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class 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.