2010/11 Taught Postgraduate Module Catalogue

MATH1035 Analysis

20 Credits Class Size: 200

Module manager: Dr M Daws (Sem 1) Professor H. G. Dales (Sem 2)
Email: mdaws@maths.leeds.ac.uk

Taught: Semesters 1 & 2 (Sep to Jun) View Timetable

Year running 2010/11

Pre-requisite qualifications

A good A-level Mathematics grade or equivalent.

Mutually Exclusive

Module replaces

MATH1031 and MATH1201

This module is approved as an Elective

Module summary

This module is an introduction to limits and their role in the calculus. It continues by looking at the continuity and differentiability of functions on the real line. As a necessary preliminary to the understanding of limits, we first clarify our ideas of (rational, real and complex) numbers, and also we look at standard methods of proof.

Objectives

On completion of this module, students should be able to:

a) deal with elementary properties of integers, rational, real numbers
b) solve algebraic problems involving complex numbers
c) use set and function notation
d) handle statements involving the quantifiers "for all" and "there exists"
e) construct proofs using mathematical induction, proof by contradiction and other common methods
f) solve simple inequalities
g) find the limits of standard kinds of sequences and functions
h) test series for convergence
i) find the radius of convergence of standard kinds of power series
j) understand, prove and apply basic results on limits, continuity and differentiability.

Syllabus

1. The main number systems: natural numbers, integers, rational numbers, real numbers.
2. Complex numbers, the Complex Plane, modulus and argument, triangle inequality, De Moivre's Theorem, n-th roots of complex numbers.
3. Sets and functions. One-one and onto functions. Inverse functions.
4. Quantifiers. Proofs and counterexamples. Proof by Mathematical Induction.
5. Rational numbers. Equivalence relations.
6. Solution of inequalities.
7. Sequences: The idea of a sequence in R. Rules for limits of sums and products of sequences; the squeeze rule. Monotone sequences. Increasing sequence either converges or tends to infinity.
8. Definition of a series, partial sums, convergence of a series. Harmonic and geometric series. Elementary properties of series. Tests for convergence and divergence. Examples. Alternating series, absolute convergence. Power series.
9. Functions: Limits and continuity of real-valued functions at a point via the - definition; condition involving convergent sequences. One-sided limits. Properties of continuous functions; sums, products, compositions of continuous functions are continuous. The boundedness property of continuous functions on a closed interval, the intermediate value theorem, the interval theorem.
10. Differentiability: Informal discussion of derivative in terms of gradients and velocities. Formal definition of the derivative. Basic rules for differentiation (assuming rules for limits); the chain rule. Applications to maxima and minima. Rolle's theorem and the mean value theorem. l'Hôpital's rule for indeterminate forms.

Teaching Methods

Opportunities for Formative Feedback

10 Problems sheets at two week intervals.

Methods of Assessment

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading List

The reading list is available from the Library website

Last updated: 01/04/2011

Errors, omissions, failed links etc should be notified to the Catalogue Team