Co-requisite: MAST4006 (Mathematical Methods 1)
OverviewTopics covered will include:
Real Numbers: Rational and real numbers, absolute value and metric structure on the real numbers, induction, countability and uncountability, infimum and supremum.
Limits of Sequences: Sequences, definition of convergence, epsilon terminology, uniqueness, algebra of limits, comparison principles, standard limits, subsequences and non-existence of limits, convergence to infinity.
Completeness Properties: Cantor's Intersection Theorem, limit points, Bolzano-Weierstrass theorem, Cauchy sequences.
Continuity of Functions: Functions and basic definitions, limits of functions, continuity and epsilon terminology, sequential continuity, Intermediate Value Theorem.
Differentiation: Definition of the derivative, product rule, quotient rule and chain rule, derivatives and local properties, Mean Value Theorem, L'Hospital's Rule.
Taylor Approximation: Taylor's Theorem, remainder term, Taylor series, standard examples, O and o notation, limits using Taylor series.
This module appears in:
Method of assessment
80% examination and 20% coursework.
There is no essential reading or core text. Recommended background reading includes:
• D. Brannan: A first course in mathematical analysis. 2006.
• F. Hart: Guide to analysis. 2001.
• A. Mattuck: Introduction to analysis. 2013.
Additional reading includes:
• L. Alcock: How to think about analysis. 2014.
On successfully completing the module students will be able to:
1 demonstrate knowledge of the underlying concepts and principles associated with continuity and differentiability of real functions;
2 demonstrate the capability to make sound judgements in accordance with the basic theories and concepts in the following areas, whilst demonstrating a reasonable level of skill in calculation and manipulation of the material: real numbers, limits of sequences, completeness properties, continuity of functions, differentiation, Taylor approximation;
3 apply the underlying concepts and principles associated with continuity and differentiablility in several well-defined contexts, showing an ability to evaluate the appropriateness of different approaches to solving problems in this area.
The intended generic learning outcomes.
On successfully completing the module students will be able to demonstrate an increased ability to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make use of information technology skills such as online resources (moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation.