The University of Southampton

MATH1059 Calculus

Module Overview

This module provides a bridge between A-level mathematics and university mathematics. Some of the material will be similar to that in A-level Maths and Further Maths but will be treated in more depth, and some of the material will be new. Topics of study include functions, limits, continuity, differentiation, integration and ordinary differential equations.

Aims and Objectives

Module Aims

This module aims to introduce the student to the main ideas and techniques of differential and integral calculus, and elementary ordinary differential equations.

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • Be able to accurately handle inequalities for real numbers
  • Be able to understand the definition of limit, and be able to use the algebra of limits, l'Hopital's rule, etc., to determine limits of simple expressions
  • Be able to understand the definitions of continuity and differentiability, and be able to differentiate continuous functions.
  • Be able to state and apply the Intermediate Value Theorem, Rolle's Theorem and the Mean Value Theorem
  • Be able to understand the concept of integration, and be able to apply different methods of integration to find areas under curves
  • Be able to solve a range of first and second order ordinary differential equations


Functions • Domain, range. • Inverse functions. • Injection, surjection, bijection. Limits • Informal and formal concepts of limit. • Examples of definition of limit in terms of epsilon and delta: linear and scalar cases. • Algebra of limits. • Definition of continuity; sums, products and compositions of continuous functions. • Intermediate Value Theorem. Differentiation • Slopes and tangent lines, formal definition in terms of a limit. • Rules of differentiation: product rule, quotient rule and chain rule. • Differentiation of trigonometric and hyperbolic trigonometric functions, inverse trigonometric functions, the exponential and logarithmic functions. • Rolles' Theorem and the Mean Value Theorem. • L’Hôpital’s rule. • Maxima and minima, curve sketching. • Taylor series. Integration • The concept of integration via Riemann sums. • Fundamental Theorem of Calculus. • Methods of integration: substitution, parts, partial fractions. • Improper integrals. Differential Equations • First order differential equations: separable, linear, homogeneous, Bernoulli, Clairaut. • Second order constant coefficient differential equations: complementary functions and particular integrals.

Learning and Teaching

Teaching and learning methods

Lectures, problem classes, workshops, private study

Supervised time in studio/workshop6
Wider reading or practice10
Follow-up work24
Preparation for scheduled sessions12
Completion of assessment task20
Total study time150

Resources & Reading list

ROBINSON J.C.. An Introduction to Ordinary Differential Equations. 

SPIVAK M.. Calculus. 

ADAMS R.A.. Calculus - A complete course. 

AYRES F. MENDELSON E.. Calculus, Schaum's outline series. 



Class Test


MethodPercentage contribution
Class Test 10%
Coursework 20%
Written exam  (2 hours) 70%


MethodPercentage contribution
Written exam 100%

Repeat Information

Repeat type: Internal & External

Linked modules


To study this module, you will need to also study the following module(s):

MATH1048Linear Algebra I
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