Module Catalogues, Xi'an Jiaotong-Liverpool University

Module Code: MTH117
Module Title: Analysis 1
Module Level: Level 1
Module Credits: 5.00
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: 6/5/2020 8:44:43 PM