Module Identifier MA10020  
Academic Year 2003/2004  
Co-ordinator Dr T McDonough  
Semester Semester 1  
Other staff Dr David M Binding, Dr R S Jones  
Pre-Requisite A-level Mathematics or equivalent.  
Mutually Exclusive MA12110, MA12510, MA12610, MA13010, MA13510, MA13610  
Course delivery Other   Workshop. 4 x 1 hour workshops (including test)  
  Lecture   40 Hours 40 x 1 hour lectures  
  Seminars / Tutorials   10 Hours 10 x 1 hour tutorials  
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  75%
Semester Assessment Continuous Assessment:  25%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

On completion of this module, a student should be able to:

Brief description

This module covers the algebra and calculus which are fundamental to the development of mathematics.


To introduce students to the ideas of algebra through the study of complex numbers and polynomials; to establish a clear understanding of the ideas of limit and derivative; to develop technical facility in calculations involving limits and derivatives and to develop techniques for determining definite and indefinite integrals.


1. SETS AND MAPPINGS: Introduction to number systems and mappings.
2. FINITE SUMS: The Binomial Theorem, arithmetic and geometric series. The Principle of Mathematical Induction.   
3. COMPLEX NUMBERS: Geometric interpretation. DeMoivre's Theorem.
4. POLYNOMIALS: The Division Algorithm and the Remainder Theorem. Symmetric functions. Relations between roots of a polynomial and its coefficients.
5. FUNCTIONS OF A REAL VARIABLE: Graphs of elementary functions (polynomial, trigonometric, exponential, logarithm, absolute value, integer part). Periodic functions, even and odd functions. Operations on functions: addition, multiplication, division, composition.
6. LIMITS AND CONTINUITY: Limit notation. Rules for manipulation of limits. Sandwich theorem for limits, applications. Definition of continuity at a point in terms of limits. Continuity of sum, product, quotient and composite of continuous functions. Intermediate - value theorem.
7. DIFFERENTIATION: Fermat's difference quotient (f(x)-f(a))/(x-a). Definition of the derivative of f(x) at a point. Geometric significance of the derivative. Differentiation from first principles of some elementary functions. Continuity of differentiable function; examples of continuous nondifferentiable functions. Rules for differentiation. Examples on differentiation, including logarithmic differentiation. Second-order derivatives.
8. INVERSE FUNCTIONS: Definition. Trigonometric and polynomial examples. Differentiation of elementary inverse functions.
9. LOCAL MAXIMA AND MINIMA, CURVE SKETCHING: Locating the critical points of a function. Using the first derivative test to determine local maxima and minima. Points of inflexion. Graphs of rational functions, vertical asymptotes, horizontal asymptotes.
10. INTEGRATION: The Fundamental theorem of the integral calculus. Linearity properties of integration. Indefinite integrals. Methods of integration: integration by substitution, integration by parts. Definition of log x as an integral. Properties of the log function from the properties of the integral. The exponential function as the inverse of the log function. The hyperbolic functions. Integration of rational functions, use of partial fractions.

Reading Lists

** Recommended Text
J Stewart (2001) Calculus : concepts and contexts 2/e. Brooks/Cole 0534377181
M W Liebeck (2000) A concise introduction to pure mathematics CRC Press 1584881933
** Supplementary Text
R A Adams (1999) Calculus - a complete course 4/e. Addison-Wesley 0201396076
R L Finney and G B Thomas, [FT] (1994) Calculus 2nd. Addison-Wesley 0201549778
K E Hirst (1995) Numbers, sequences and series Arnold 0340610433
D W Jordan and P Smith (1994) Mathematical techniques: an introduction for the engineering, physical and mathematical sciences Oxford University Press 0198562683
Salas, Hille and Ergen (2003) Calculus 9th ed. Wiley 0471383759


This module is at CQFW Level 4