|Module Title||MATHEMATICAL ANALYSIS|
|Co-ordinator||Dr R J Douglas|
|Other staff||Dr R J Douglas, Dr V C Mavron, Dr J M Pearson|
|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||6 x 1 hour tutorials|
|Workshop||2 x 1 hour workshops (including test)|
|Assessment||Exam||2 Hours (written examination)||75%|
|Resit assessment||2 Hours (written examination)||100%|
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.
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.
On completion of this module, a student should be able to:
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. Establishing inequalities using 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.
6. INTEGRATION: Riemann sums. The definite integral as a limit of Riemann sums. Mean-value theorems for integrals. Fundamental theorem of the integral calculus. Reduction formulae.
** Recommended Text
R L Finney and G B Thomas. (1994) Calculus. 2nd edition. Addison-Wesley
** Supplementary Text
J Stewart. (1999) Calculus. 4th edition. Brooks/Cole
K E Hirst. Numbers, Sequences and Series. Arnold
R Adams. Calculus - a Complete Course. Addison-Wesley
R Haggarty. Fundamentals of Mathematical Analysis. Addison-Wesley
R G Bartle & D R Sherbert. Introduction to Real Analysis. Wiley