Module Catalogues, Xi'an Jiaotong-Liverpool University   
 
Module Code: MTH117
Module Title: Analysis 1
Module Level: Level 1
Module Credits: 5.00
Academic Year: 2019/20
Semester: SEM1
Originating Department: Mathematical Sciences
Pre-requisites: MTH002ORMTH004ORMTH008ORMTH014 AND MTH013 AND MTH013ORMTH019ORMTH023ORMTH027 MTH008
   
Aims
To provide a basic understanding of the principles of Real Analysis.
Learning outcomes 
A State definitions and theorems in real analysis;

B Present proofs of the main theorems;

C Apply these definitions and theorems to simple examples;

D Construct their own proofs of simple unseen results.

Method of teaching and learning 
This module will be delivered by a combination of formal lectures and tutorials.
Syllabus 

The real numbers: ordered field axioms, Cantor’s form of the completeness axiom, countable and uncountable sets, supremum, the Property of the Supremum.


Elementary functions: polynomials and rational functions, exponential and logarithmic functions, trigonometric functions.


Topology of the Real Line: accumulation points, open and closed sets, compact sets, Bolzano-Weierstrass’s Theorem.


Limits and continuity: definition of limit, algebra of limits, definition of continuity, continuity of the elementary functions, algebra of continuous functions, classification of discontinuities.


Properties of the continuous functions: extrema, relative extrema, Weierstrass’s Theorem, Bolzano’s Theorem, Intermediate Value Theorem, monotone functions.


Sequences of real numbers: definition, limits, algebra of limits, monotone sequences, subsequences, oscillation limits, upper and lower limits, Cauchy sequences.


Series of real numbers: definition, convergence, convergence test, absolute convergence, alternating sequences.

Delivery Hours  
Lectures Seminars Tutorials Lab/Prcaticals Fieldwork / Placement Other(Private study) Total
Hours/Semester 39    13      98  150 

Assessment

Sequence Method % of Final Mark
1 Formal Examination 70.00
2 Class Test1 15.00
4 Class Test2 15.00

Module Catalogue generated from SITS CUT-OFF: 8/20/2019 6:20:53 PM