[The University of Leicester]

Department of Mathematics



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MA2101 Real Analysis

MA2101 Real Analysis

Credits: 10 Convenor: Dr. A. E. Henke Semester: 1 (weeks 1-6)
Prerequisites: essential: MA1151
Assessment: Individual and group coursework: 20% One and a half hour hour exam: 80%
Lectures: 18 Problem Classes: 5
Tutorials: none Private Study: 47
Labs: none Seminars: none
Project: none Other: none
Surgeries: 5 Total: 75

Subject Knowledge

Aims

The main aim of this course is to develop knowledge in basic mathematical analysis, continuing the study in MA1151.

Learning Outcomes

To know the definitions of, and understand the key concepts introduced in, this module.

To understand, reconstruct and apply the main results and proofs covered in the module.

To know the definition of convergence for infinite series, and test for convergence using standard tests.

To know the formal definitions of differentiation and Riemann integration.

Basic mathematical analysis.

Methods

Class sessions and surgeries together with some handouts.

Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the January exam period. The 20% coursework contribution will be determined by students' solutions to four sets of work, one of which will be done in groups. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.

Subject Skills

Aims

To provide students with team working skills and develop written communication skills and problem solving skills.

Learning Outcomes

Students will be able to use the techniques taught within the module to solve problems, and be able to present arguments and solutions in a coherent and logical form. The ability to synthesise and build upon existing information is achieved.

Methods

Class sessions, surgeries.

Assessment

Marked problem sheets, team work project, exam.

Explanation of Pre-requisites

This module continues the study, started in MA1151, of real analysis and its applications; it thus builds directly upon the material studied in MA1151.

Course Description

This module deals with differentiation and integration of real-valued functions. It puts these notions on a firm footing with rigorous definitions, and it provides tools to handle complicated functions such as $sin(x)$, $cos(x)$ or $e^x$ which are used extensively in many parts of mathematics and its applications. The approach is based on approximating such functions by easier ones. For example, it is easy to differentiate or integrate polynomials. Thus, if we can approximate a function by polynomials, we will have a chance to handle this function as well. Such an approximation is based on Taylor's theorem which allows us to express some functions as convergent power series, i.e. certain sequences of polynomials.

Therefore, much of the course is devoted to studying sequences and series and to develop criteria for their convergence. Applying the results to problems of differentation and integration one obtains both a rigorous theory and useful practical tools.

Syllabus

Differentiation:

Local maximum and minimum, Rolle's theorem, Mean Value Theorem, differentiation of power series, Taylor's series and theorem.


Sequences:

Subsequences, Bolzano-Weierstrass Theorem, Cauchy sequences.


Series:

Infinite series, geometric series, harmonic series, comparison test, ratio test, alternating series, conditional convergence, sums and products of series, power series, radius of convergence.


Integration:

Dissection of an interval, upper and lower sums, refinement, upper and lower integrals, integrable functions, fundamental theorem of calculus, limitations of the theorem of Riemann integration.

Reading list

Background:

R. Haggarty, Fundamentals of Mathematical Analysis, Adison Wesley.

M. Spivak, Calculus, Benjamin-Cummings.

D. Stirling, Mathematical Analysis: a fundamental and straightforward approach, Ellis Horwood.

Resources

Problem sheets and additional handouts.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA2102 Linear Algebra Up: ModuleGuide03-04 Previous: MA2081 Methods of Applied Mathematics II

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Last updated: 2004-02-21
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