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MC248 Further Real Analysis

MC248 Further Real Analysis

Credits: 10 Convenor: Prof. S. König Semester: 1 (weeks 1-6)
Prerequisites: essential: MC146
Assessment: Individual and group coursework: 20% One and a half hour hour exam: 80%
Lectures: 18 Problem Classes: 5
Tutorials: none Private Study: 52
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Explanation of Pre-requisites

This module continues the study, started in MC146, of real analysis and its applications; it thus builds directly upon the material studied in MC146.

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.

Objectives

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.

Transferable Skills

Basic mathematical analysis. Groupwork.

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.

Details of 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.

Next: MC254 Algebra I Up: Year 2 Previous: MC243 Aspects of Linear Analysis

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Last updated: 2001-09-20
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