Module Content

Revision of Calculus
Differentiation: Rules for sum, difference, product and quotient; Function of a function; Standard derivatives; Parametric and implicit equations; Higher derivatives; Maxima and Minima
Integration: Appreciation of the techniques of integration by substitution, parts and partial fractions. Applications to area and volumes.
Functions
Basic properties of circular, exponential and hyperbolic functions and their inverses. Parametric representation of functions.
Power Series
Intuitive idea of a convergent infinite series. Taylor-Maclaurin series. Series for the standard functions. Binomial theorem. Approximation.
Complex Numbers
Definition, sum, difference, product and quotient. Argand diagram. Polar form; products and quotients in polar form. De Moivre’s theorem. Exponential form and Euler’s Formula. Roots of equations.
Partial Differentiation
Functions of two or more variables. Partial derivatives. Taylor series for functions of two variables. Total differential. Application to errors and small changes. Change of variables, the chain rule. Stationary points of functions of two variables, local maxima, minima and saddle points.
Vectors
Vectors and scalars; laws of vector algebra. Unit vectors, components of a vector. Scalar and vector products. Triple vector products. Intro to vector calculus: grad, div and curl. 
Multi-Dimensional Integrals
Definitions and evaluation of double and triple integrals. Use of plane polar, spherical polar and cylindrical polar co-ordinate systems. Applications to areas, volumes, centres of mass etc.

Teaching Methods

The module will be delivered on campus, with weekly lecture and tutorial sessions. 

Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding. 

Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials. 

Assessment Methods

The module is assessed through a Portfolio and an examination.