Module Identifier MA11110
Module Title MATHEMATICAL ANALYSIS
Co-ordinator Dr Robert J Douglas
Semester Semester 2
Other staff Ms Brenda M Hughes
Pre-Requisite MA10020
Mutually Exclusive May not be taken at the same time as any of MA12010 to MA13510.
Course delivery Lecture   20 x 1 hour lectures
Seminars / Tutorials   5 x 1 hour tutorials
Other   Workshop. 2 x 1 hour workshops (including test)
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  75%
Semester Assessment Continuous Assessment:  25%
Supplementary Assessment2 Hours (written examination)  100%

#### Learning outcomes

On completion of this module, a student should be able to:
• determine solution sets of elementary inequalities;
• determine whether or not a set of real numbers is bounded;
• determine the supremum and infimum of bounded sets;
• describe the notion of a sequence of real numbers and determine whether sequences are convergent or divergent;
• apply the standard theorems on convergence of sequences;
• manipulate sequences defined by recurrence relationships;
• use the basic tests for convergence of series;
• compute the higher derivatives of certain functions and use Leibnitz''''s theorem;
• state and use the mean-value theorem of the differential calculus, Taylor''''s theorem and Maclaurin''''s theorem.

#### Brief description

A first course in Mathematical Analysis aims to tackle some of the issues which are glossed over in the development of calculus. The central concepts of limit and continuity will be introduced and used to prove rigorously some of the fundamental theorems in analysis. These ideas play a basic part in the subsequent development of mathematics.

#### Aims

This module aims to tackle some of the issues which are glossed over in the development of the calculus. The central concepts of limit and continuity will be introduced and used to prove rigorously some of the fundamental theorems in analysis. The theoretical aspects of the subject will be developed in conjunction with the techniques required to solve problems.

#### Content

1. INEQUALITIES: Solution sets for rational inequalities.
2. BOUNDED SETS: Upper bound, lower bound, infimum, supremum. Completeness axiom for the real numbers.
3. SEQUENCES: Limit of a convergent sequence of real numbers. Formal derivation of some limit theorems. The sandwich theorem. Sequences defined by recurrence relationships. Increasing and decreasing sequences and related convergence theorems. Boundedness of convergent sequences. Subsequences.
4. APPLICATIONS OF THE DIFFERENTIAL CALCULUS: Rolle's theorem. Mean-value theorem of the differential calculus.   L'Hopital's rule. Leibnitz's theorem. Taylor's theorem, Maclaurin's theorem.
5. INFINITE SERIES: Partial sums. Convergence of infinite series. Examples of convergent and divergent series, including geometric series. Tests for convergence of series of positive terms: comparison test, ratio test, integral test.

Books
** Recommended Text
Weir, M D, Haas, J and Giordano, F R (2005) Thomas' Calculus Addison Wesley 0321243358
J Stewart (2001) Calculus: concepts and contexts 2nd edition. Brooks/Cole 0534377181
** Supplementary Text
R Haggarty (1993) Fundamentals of Mathematical Analysis 2nd. Addison-Wesley 0201631970