# REAL ANALYSIS 1 - 2017/8

Module code: MAT1032

Module provider

Mathematics

ZELIK S Prof (Maths)

Number of Credits

15

ECTS Credits

7.5

Framework

FHEQ Level 4

JACs code

G120

Module cap (Maximum number of students)

N/A

Module Availability

Semester 1

Independent Study Hours: 101

Lecture Hours: 33

Seminar Hours: 5

Tutorial Hours: 11

Assessment pattern

Assessment type Unit of assessment Weighting
Examination EXAMINATION 75
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 25

Alternative Assessment

N/A

Prerequisites / Co-requisites

None.

Module overview

This module is an introduction to analysis, which is the branch of mathematics that rigorously studies functions, continuity and limit processes, such as differentiation and integration. The module leads, among other things, to a deeper understanding of what it means for a sequence or series to converge. Tools such as convergence tests are presented and their validity proved, and the rigorous use of definitions and logic play a central role.   This course lays the foundations for the Year 2 module in Real Analysis 2 (MAT2004) in particular, and, more generally, underpins other modules where a culture of rigorous proof exists.

Module aims

Introduce students to quantifiers, logical statements, countability, suprema, maps, sequences and series.

Enable students to determine limits of sequences and to test and prove the convergence of series and sequences.

Illustrate the application of various techniques for solving frequently encountered problems in analysis.

Learning outcomes

Attributes Developed
1 Demonstrate understanding of the real numbers and the role of completeness in the existence of limits and solutions to equations. K
2 Interpret and apply quantifiers in mathematical statements, and quote and apply basic theorems in analysis. KCT
3 Calculate limits of sequences and (power) series, and prove/disprove convergence using the definitions KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Module content

Indicative content includes:

The axioms of real numbers. Denseness of rational and irrational numbers. Maximum, minimum, supremum and infimum of sets, sequences and functions. The triangle inequality.

Natural induction, set notation, cardinalities of sets (in particular the rationals and reals)

Axiom of Completeness, and its consequences for the existence of limits. Role of quantifiers in stating and verifying mathematical definitions.

Sequences and convergence, and their properties. Boundedness, Cauchy sequences, subsequences and the Theorem of Bolzano-Weierstrass.

Infinite series, convergence and absolute convergence. Convergence tests. Power series, radius and region of convergence.

Continuity of real-valued functions

Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

A detailed introduction to the real numbers, sequences, series and basic ideas of convergence
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.
(every second week) 1 x 1 hour seminar for guided discussion of solutions to problem sheets provided to and worked on by students in advance

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 two hour examination (two of three best answers contribute to exam mark, with Question 1 compulsory) at the end of Semester 1; worth 75% module mark.

·         One in-semester test; worth 25% module mark.

Formative assessment and feedback

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