Aims

To lay the foundations of calculus and differential equations for more advanced mathematical study. Students will compute derivatives and integrals using standard techniques. They will learn to solve simple first and second order ordinary differential equations

Module summary

Virtually every branch of mathematics, statistics, and physics can be developed only from a firm foundation. These skills form the toolkit required for further study. A clear understanding and appreciation of many fundamental topics is required, primarily, those of algebra and calculus. This module concentrates on developing further the techniques of calculus the students have already seen as part of an A-level or equivalent qualification. The techniques developed in calculus are useful when constructing mathematical models of phenomena in the real world. Many such models are formulated in terms of ordinary differential equations, and this module introduces the methods that are needed to solve problems of this type.

Outline Of Syllabus

Definition of derivatives and derivatives of elementary functions from first principle

Continuity and differentiability

Product, quotient and chain rules

Implicit differentiation

Review of inverse of a function, standard examples and derivatives of inverses

Hyperbolic trigonometric functions and derivatives

Maclaurin and Taylor Series

Problems of convergence of power series and series in general

 Integral as area under a curve, as the limit of series

Statement of Fundamental Theorem of Calculus

Integration by parts, by substitution

Standard integrals

Integration by reduction

First-order ODEs: separable equations, homogeneous equations, integrating factor. A brief introduction to isoclines. 

Second-order ODEs: homogeneous equations with constant coefficients, particular integrals for inhomogeneous equations, method of reduction of order.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.