## Mathematics I

### Module aims

This module introduces mathematics as a logical and structured discipline, focussing on providing the mathematical knowledge and skills required for their First Year Civil Engineering programme.

It provides a basis for the more advanced mathematical techniques which are required in subsequent years of the programme, with an emphasis on application of mathematics to the solution of engineering problems.

### Learning outcomes

· Characterise functions and apply differential calculus to the solution of optimisation, related rates and approximation problems.

· Manipulate complex numbers in algebraic, polar and exponential forms. Use MacLaurin and Taylor series to establish approximations to functions.

· Integrate functions using a variety of techniques and apply them to the determination of areas, volumes and average values of functions.

· Find the inverse of matrices by the Gauss-Jordan method and the solution of linear algebraic equations by Gaussian elimination and LU factorisation.

· Solve certain first and second order ordinary differential equations.

### Module syllabus

Functions of One Variable: Definitions, Limits, Continuity, Differentiation, Curve Sketching & Applications Complex Numbers Series:

Definitions, Taylor, Maclaurin & Binomial Series Integration: Techniques, Definite Integral, Improper Integral & Applications Parametric Functions, Polar Coordinates & Hyperbolic Functions Functions of More Than One Variable:

Derivatives, Directional Derivative, Gradient & Applications Matrices & Vectors:

Basic Operations, Geometric Interpretation, Rank, Systems of Linear Equations, Gaussian Elimination Determinants, Cramer’s Rule, Inverse of a Matrix & Gauss-Jordan Elimination Eigenvalues and Eigenvectors, Diagonalisation and Powers of Matrices Vector Algebra: Scalar & Vector Products, Equations of Lines & Planes First order ODEs: Formation, Linear ODEs and Integration Factors, Exact ODEs & Applications Second Order ODEs: Double and Complex Roots, Undetermined Coefficients

### Assessments

Assessment Information is provided separately in Blackboard Learn.