University of Wales, Bangor - School of Informatics Mathematics Part 1 Modules

Lecturer: Dr Chris Wensley

Recommended book:
Calculus and Analytic Geometry, 9th Ed, G B Thomas & R L Finney, Addison Wesley, 1996, ISBN 0201400154.

Learning outcomes:

This module aims to develop an intuition for limiting processes and an understanding of epsilon-delta arguments concerning these processes.
It provides an introduction to the more theoretical aspects of analysis, balancing the technique approach of the mathematical methods modules.
Nevertheless, these difficult ideas will be illustrated with many standard examples, and the Maple computer algebra system will be used to provide extensive backup to the module.
Success in achieving these aims is to be demonstrated through a series of assessments as indicated below.

Key skills:

• Ability to think and argue, particularly using epsilon-delta proofs for limits.
• Ability to manipulate inequalities, and apply standard algorithms for testing convergence.
• Ability to read and understand textbook material.
• Ability to produce computer algebra worksheets illustrating problems in analysis.
• Understand the development of mathematical ideas with particular reference to Taylor's theorem.

Pre-requisites: IMM1001

Syllabus

Limits
Limits of real functions; the floor and ceiling functions; target value problems.
Sum, product and quotient and de L'Hopital's rules. Standard types of example.

Sequences
Sequences defined by a function, by iteration, or by recurrence relation.
Limits and convergence of sequences. Experiments with Quattro and Maple worksheets.
Solution of second-order, linear recurrence relations.
Convergence of the sum, product and quotient of convergent sequences.

Series
Series as sequences of partial sums. Geometric series.
Tests for convergence of series of positive terms: ratio, comparison; limit comparison; integral.
Film Space-filling curves. Area of a snowflake curve.
Alternating series test. Absolutely convergent series. Radius of convergence for power series.

Approximation of functions
Rules for manipulating inequalities. Algorithm for solving rational inequalities.
Revision of Taylor and Maclaurin series. Derivative and integral forms of the remainder term.

Continuity & Differentiation
Continuity and properties of continuous functions. Sandwich composite and gluing rules.
Statement of the Intermediate value, Rolle and mean value theorems.
Location of roots of an equation.
Functions acting on sets.

Functions of 2 variables
Limit of a function as (x,y) tends to (a,b) along a curve. Sandwich rule.

Assessment

• Continuous assessment: 40%
• 1½ hour closed book, end of semester exam: 60%

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