Overview

Indicative content includes:

Differentiation: Revision of notation for sets and functions. Revision of basic algebra techniques. The limit of a real function at a point. Derivative as gradient: tangent lines. Rules of differentiation. Polynomial, exponential, logarithmic, and inverse functions. Local maxima, minima, points of inflection. Using MATLAB to sketch graphs of functions. Parametric curves; polar coordinates. Solution of equations by iteration

Integration: Integration as anti-differentiation. The area under a curve. Integration by parts and by substitution. Methods of numerical integration. Mean and root-mean-square values. The method of partial fractions. Integrals of the form f’(x)/f(x) and f’(x).f(x). Distance, velocity and acceleration. Parametric curves: arc length and area. Maclaurin and Taylor series expansions. Arithmetic with Maclaurin series.

Number Systems: Integers, rationals and real numbers. Fractions ↔ infinite decimal expansions. j = √( -1), solving quadratic equations. Complex arithmetic; Argand diagram. Revision of trigonometric functions. Complex functions: bilinear; exp; log. exp(jθ) = cos(θ) + j.sin(θ). De Moivre’s theorem: n-th roots

Functions of two variables: Examples of real functions of two variables. Using software to sketch surfaces. Partial differentiation and tangent planes. Maxima, minima and saddle points. Solution of exact differential equations. Maclaurin series for f(x,y).

Probability and Statistics: Arrangement of data; histograms; mean; mode; and median. Dispersion; range; standard deviation. Normal distribution; standardized normal curve. ; Probability; Independence; Mutually exclusive events; Probability density functions; The binomial distribution; The Poisson distribution; The Gaussian distribution.

Matrices: Matrix definition; Operations on Matrices; Determinants; Matrix inversion; Solutions of linear equations using matrices;

Vectors and vector fields: Vector definition; Operations on vectors. Vectors in 2D and 3D; Representation of lines and planes using vectors; Scalar and vector products of two vectors; Angle between two vectors; Scalar fields; Vector fields; Gradient of a scalar field

Assessment Strategy

-threshold -Equivalent to 40%. Uses key areas of theory or knowledge to meet the Learning Outcomes of the module. Is able to formulate an appropriate solution to accurately solve tasks and questions. Can identify individual aspects, but lacks an awareness of links between them and the wider contexts. Outputs can be understood, but lack structure and/or coherence.

-good -Equivalent to the range 60%-69%. Is able to analyse a task or problem to decide which aspects of theory and knowledge to apply. Solutions are of a workable quality, demonstrating understanding of underlying principles. Major themes can be linked appropriately but may not be able to extend this to individual aspects. Outputs are readily understood, with an appropriate structure but may lack sophistication.

-excellent -Equivalent to the range 70%+. Assemble critically evaluated, relevant areas of knowledge and theory to construct professional-level solutions to tasks and questions presented. Is able to cross-link themes and aspects to draw considered conclusions. Presents outputs in a cohesive, accurate, and efficient manner.

Learning Outcomes

• Apply basic operations in vector algebra (Cartesian and geometric representations) to represent lines and planes, calculate the gradient of a scalar field using partial derivatives.

• Apply the basic rules and techniques of **differential** calculus and its application in engineering.

• Apply the basic rules and techniques of **integral** calculus and its application in engineering.

• Apply theory of probability and statistics to calculate averages, data spread and the probability of different events.

• Demonstrate the basics, rules and techniques for differential equation and partial differentiation.

• Demonstrate the basics, rules and techniques of complex number algebra and its application in engineering.

• Use basic operations of matrix algebra, determinants and their application in solving systems of linear equations.

• Use of software packages for statistical, probabilistic and matrix calculations.