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MODULE DESCRIPTION

Module Code: MA0124

Module Title: Calculus Methods II

Department Responsible: MATHS

Module Tutor: Professor V I Burenkov

Number ofCredits: 10 Level: 1

Semester: Spring Number ofSemesters: 1

Schemes of Study For WhichThis Module Is Compulsory:

Schemes of Study For WhichThis Module Is Optional:

Title of Scheme: JACS Code: Years: Title of Scheme: JACS Code: Years:Mathematics G100 1Mathematics and itsApplications

G120/111 1

Mathematics, OperationalResearch and Statistics

G991/990 1

Computing and Mathematics GG14 1Mathematics and Physics FG31 1BA Joint Honours GR11, GR12,

GV15, GW13,GX13,QG51,

VG61

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Module to be offered on a Free-Standing basis? Yes

Precursor Modules: Code: Title:MA0121 Calculus Methods I

AIMS OF THE MODULE

1. To introduce students to second-order linear differential equations.2. To develop proficiency in using manipulative techniques to obtain solutions of second-order differential equations with

constant coefficients.3. To make students aware of the issues of existence and uniqueness for solutions of linear differential equations.4. To allow students to appreciate the geometrical interpretation of integrals and their definition using Riemann sums.5. To apply the techniques of integration for a range of applications within mathematics.

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CATALOGUE ENTRY

The first part of the module is concerned with the solution of second-order differential equations, and thus continuesdirectly on from the work completed in Calculus Methods I for first-order differential equations. Manipulative techniques willbe used to determine solutions of second-order differential equations for cases where the equation takes a specific andrelatively simple form. There will also be some general discussion about the circumstances under which it is possible toknow that there is a solution of a differential equation, even if a simple mathematical formula for the solution cannot beobtained.

The second part of the module reconsiders the nature of definite integrals. It focuses on the geometrical interpretation ofintegrals, by carefully considering how they are connected with the behaviour of sums that approximate areas undergraphs. Definitions and interpretations will also be given for improper integrals that involve infinite ranges of integration orfunctions with singularites. A variety of applications of integration will be made, including the use of improper integrals totest whether or not an infinite series is convergent.

LEARNING OUTCOMES OF THE MODULE

Knowledge and Understanding:

On completion of the module a student should be able to:

1. Obtain solutions of second-order differential equations with constant coefficients.

2. Understand the general structure of solutions to second-order linear differential equations and appreciate thesignificance of existence and uniqueness for such solutions.

3. Understand the geometrical interpretation of definite integration and its relationship to Riemann sums, areas and anti-derivatives.

4. Use integration techniques to calculate integrals that appear in a variety mathematical applications, including improperintegrals.

Skills:

Solution methods for linear second-order differential equations. An understanding of the nature of integration that underliesthe capability to apply manipulative integration techniques.

Transferable Skills:

The ability to make meaningful use of integration and to determine an appropriate solution of a differential equationare both examples of mathematical skills that can be deployed to tackle a wide variety of problems that are found inapplications.

METHODS OF TEACHING AND LEARNING:

27 hours lectures and examples lectures5 hours tutorial classes.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions fortutorial classes.

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ASSESSMENT:

Formative assessment is by means of regular tutorial exercises. Feedback to students on their solutions and theirprogress towards learning outcomes is provided during lectures and tutorial classes.

The in-course element of summative assessment is based on selected problems on the tutorial sheets.

The major component of summative assessment is the written examination at the end of the module. This gives studentsthe opportunity to demonstrate their overall achievement of learning outcomes. It also allows them to give evidence of thehigher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight. Section A contains a number of compulsory questions of variablelength but normally short. Section B has a choice of two from three equally weighted questions.

METHOD(S) OF SUMMATIVE ASSESSMENT:

Written Examination: Percentage Contribution to the Module Assessment: 85%

Semester in which Written Examination is to be Scheduled : Spring

Duration of Examination: 2 hours

In-course Assessment: Percentage Contribution to the Module Assessment: 15% Comprising:

Coursework: 15%

SYLLABUS CONTENT:

Second order differential equations with constant Coefficients• Homogeneous equations and non-homogeneous equations. Characteristic equation. General solutions and particular

solutions.• Non-homogeneous equations with right-hand sides involving combinations of polynomial, exponential and trigonometric

functions.

General second order linear differential equations• Structure of general solutions for homogeneous and non-homogeneous cases. The Wronskian and finding solutions by

the variation of parameters.• Statement of existence and uniqueness theorem for the initial value problem. Generalization to n-th order.

Definite Integrals• Examples of Riemann sums and the Riemann integral. Geometrical meaning. Existence of Riemann integral for a

piecewise continuous function (without proof).• Fundamental theorem of calculus. Differentiation of an integral with a variable upper limit. Properties of definite

integrals: substitutions and integration by parts.• Applications of integration. Arc lengths, areas and surface areas, volumes, averaging.

Improper integrals• Definitions and basic properties of improper integrals. Comparison test for the convergence of improper integrals of non-

negative functions. Integral test for the convergence of series.

INDICATIVE READING LIST

Finney R L and Thomas G B, Calculus, (Addison-Wesley)Edwards C H and Penney D E, Calculus (6th Ed) (Prentice Hill 2002)Stewart J, Calculus (5th Ed) (Thompson 2003)