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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. One of the pre-requisites for MATH1049, MATH1057, MATH1058, MATH1060 and MATH3090

Aims and Objectives

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

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

Resources & Reading list

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

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

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

SPIVAK M.. Calculus. 


Assessment Strategy

The assessment for this module is as follows: - 1 piece of formative assessment (class test) that does not carry a weighting, but must be passed at 80% or above in order to pass the module. - 3 pieces of summative assessment (a written exam, another class test and a piece of coursework). - The referral assessment for this module consists of 1 written exam with 100% weighting.


Class Test


MethodPercentage contribution
Coursework 40%
Written assessment 60%


MethodPercentage contribution
Written assessment 100%

Repeat Information

Repeat type: Internal & External

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