## MATH1050 Calculus and Mathematical Analysis

### 10 creditsClass Size: 250

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

A level Mathematics, or equivalent.

This module is approved as an Elective

### Objectives

To continue the study of Differential and Integral Calculus with some revision of A-level work, in order to provide a uniform background knowledge of the subject, and then to introduce some of the basic concepts of Mathematical Analysis. On completion of this module, students should be able to: (a) Calculate the derivatives and integrals of elementary functions. (b) Do arithmetic calculations with complex numbers, including calculation of nth roots. (c) Calculate limits of simple sequences; (d) Test series for convergence using standard tests. (e) Compute Taylor series. (f) Calculate partial derivatives of any order.

### Syllabus

Because A-level and other entry courses differ in their syllabuses, this module revises differential and integral calculus before obtaining further results. There is an extensive study of complex numbers, including the definitions of elementary functions with complex values for the variable. The course contains an introduction to mathematical analysis, the subject which provides the proofs for calculus, in discussions of the limit of a sequence, the sum of an infinite series and techniques to determine whether a series has a sum. Topics covered include: 1. Differentiation: Revision of methods of differentiation. 2. Hyperbolic functions and their inverses: Properties; derivatives. 3. Integration: Revision of methods of integration.

4. Complex numbers: Definition of complex numbers, De Moivre's Theorem; the logarithmic function. 5. Sequences: Definition of the limit of a sequence and calculation of limits; some basic theorems. 6. Infinite Series: Tests for convergence and absolute convergence of infinite series, radius of convergence of power series, differentiation and integration of power series. 7. Taylor's Series. 8. Partial differentiation; partial derivatives, the chain rule, change of variables.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures (22 hours); tutorials (11 hours).

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

2 hour written examination at end of semester (85%), coursework (15%).