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Module Specifications

Current Academic Year 2012 - 2013
Please note that this information is subject to change.

Module Title	Analysis I
Module Code	MS109
School	School of Mathematics
Online Module Resources
Module Co-ordinator	Semester 1: Turlough Downes Semester 2: Turlough Downes Autumn: Turlough Downes
NFQ level	8	Credit Rating	7.5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

This module introduces students to the principles and practice of mathematical analysis - the branch of formal mathematics that underpins calculus. There will be an emphasis on constructing rigorous mathematical proofs of results from analysis. Students will also learn how to use diagrams and informal ideas to develop their knowledge and skills in this area of mathematics.Students will participate in workshop style lectures and tutorials in which they reconstruct the theory and techniques of analysis using guided enquiry. They will attend review lectures on the course material. Students will submit their own work for assessment in the form of (i) homework problems and (ii) a portfolio of examples of the most important mathematical objects encountered in the course.

Learning Outcomes

1. Apply the epsilon-delta formulation of statements involving convergence of sequences and series, limits of functions (continuity and differentiability) and Riemann integration;
2. Construct rigorous arguments using the epsilon-delta formulation and be able to distinguish between rigorous and informal arguments;
3. Construct and analyse their own examples and counterexamples of mathematical objects arising in analysis;
4. Use different structured approaches to solve problems in mathematical analysis.

Workload	Full-time hours per semester
*Type*	*Hours*	*Description*
Lecturer supervised learning	24	Inquiry based study of course material
Lecture	12	Summary lecture of key concepts.
Tutorial	12	Student work on exercises with tutor support.
Independent learning time	140	Independent study of course material and work on exercise sheets.
Total Workload: 188

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Numbers and Completeness.
The natural, integer, rational and real numbers. Least upper bounds and greatest lower bounds. Completeness of the reals..

Sequences.
Introduction to sequences. Null sequences. Convergent and divergent sequences. Monotone sequences and the Weierstrass-Bolzano theorem. Cauchys criterion for convergence..

Series.
Introduction to series. Series with non-negative terms. Tests for convergence. Series with positive and negative terms and absolute convergence..

Continuity.
Limits of functions via epsilon-delta arguments. Properties of continuous functions. The intermediate value theorem and extreme values theorem..

Differentiability.
Differentiability via epsilon-delta arguments. Proof of the rules of differentiation. Rolles theorem and the mean value theorem..

Riemann Integration.
What is area? Upper and lower Riemann sums. Integrability. Integrability of continuous functions. The fundamental theorem of calculus..

Assessment Breakdown
Continuous Assessment	40%	Examination Weight	60%

Course Work Breakdown
Type	Description	% of total	Assessment Date
Portfolio	Students will maintain a portfolio of examples and counterexamples of the mathematical objects encountered in the course.	15%	Every Second Week
Assignment	Take home work.	20%	Every Second Week
Group assignment	Participation in group work in tutorials.	5%	Every Week

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
This module is category 3

Indicative Reading List