Aims: This is the second of two first year units intended to lead to confident and errorfree manipulation and use of standard mathematical relationships in the context of engineering mathematics. The unit will consolidate and extend topics met at Alevel, so that students may improve their fluency and understanding of the basic techniques required for engineering analysis.
Learning Outcomes: After taking this unit the student should be able to:
Solve certain classes of ODE, perform Laplace Transforms, determine probabilities of events and sequences of events, determine probabilities and find means and standard deviations of both discrete and continuous probability distributions, manipulate matrices, solve linear systems of linear algebraic equations using matrix methods, find eigenvalues and eigenvectors of square matrices, perform leastsquares fitting of data.
Skills: Numeracy; working independently.
Content: Ordinary differential equations: classification; reduction to first order form; linear constant coefficient. Laplace Transforms: definition; examples; transforms of derivatives; unit pulse, impulse and step function; shift theorems in s and t; solution of linear constantcoefficient ODEs, convolution theorem. Probability: trial, outcome, sample space and event; use of Venn diagrams; simple probability; complementation and addition rules; conditional probability; independent events; sampling; probability relating to sequences of events. Probability distributions: discrete and continuous distributions; mean, median, mode, percentiles and standard deviation; Poisson distribution; normal distribution. Matrices: definition; addition, multiplication and compatibility; Gaussian elimination; determinants; eigenvalues and eigenvectors. Miscellaneous topics: least squares fitting of data.
