MAT-10085 - Module Specification
School of Computer Science and Mathematics
Faculty of Natural Sciences

Programme/Approved Electives for 2026/27

Available as a Free Standing Elective

Co-requisites

None

Prerequisites

None

Barred Combinations

None

Description for 2026/27

Many real-world problems are described by calculus; for instance, the oscillation of a pendulum and the population evolution of a fish species can all be described by ordinary differential equations. This module, being a prerequisite for several modules in the second and third years, will introduce you to the basic techniques for solving differential equations and familiarize you with multivariable calculus, including partial derivatives, double integrals, and their applications while you'll simultaneously dive into the study of the infinitely small and infinitely large, and explore the rigorous foundations of Calculus. You will develop the theory via lectures, and consolidate your knowledge via practical sessions.

This module aims at deepening your understanding and appreciation of the origins of the essential techniques of elementary calculus, explaining why these methods work while also introducing you to solutions of ordinary differential equations and elements of multi-variable calculus. This will include developing important preliminary ideas such as the real numbers, functions, sequences and series, leading to the fundamental concept of a limit, and expanding these ideas to consider Taylor series, partial differentiation and double integrals and their applications.

Intended Learning Outcomes

use the basic concepts and theory to develop mathematical and logical arguments and to evaluate different approaches to problem solving: 2,3
define and identify concepts such as bounded, monotonic, or convergent sequences of real numbers and competently apply core mathematical techniques for finding the limits of sequences and series: 2
use the concept of a limit to define continuity of functions and hence develop the idea of a derivative and integral from first principles, leading to proving other differentiation and integration theorems including the fundamental theorem of calculus: 3
classify and solve several types of first-order ordinary differential equations (variable separable, linear and others which may be reduced to these): 1,4
solve first- and second-order, homogeneous linear ordinary differential equations with constant coefficients, as well as corresponding inhomogeneous ordinary differential equations with the right hand side of special form by the method of undetermined coefficients: 1,4
analyse the behaviour and convergence of number and power series and expand a function of one variable as Taylor series: 4
calculate partial derivatives, and find local maxima/minima, and restricted maxima/minima using the method of Lagrange multipliers, apply chain rule to multi-variable functions: 4
calculate double integrals and use them to find areas and volumes; change of variables under double integral, including polar coordinates: 4

Study hours

72 hours Lectures
24 hours Example Classes
12 hours class test preparation
24 hours assignment completion
12 hours examination revision
2 hours examination
152 hours private study

School Rules

None

Description of Module Assessment