The purpose of this module is to ensure the learner can visualise or describe physical systems using the basic elements of mathematics. The leaner must then be able to adapt these constructs as needed using the fundamental skills of algebra and manipulation of equations. The module will then review the essentials of differential and integral calculus needed for any engineering programme. The learner will then use expressions from differing areas of mathematics to describe Engineering systems, solve the resulting mathematical problems and interpret the results. These skills and concepts will enable the learner to use mathematics fluently during their career as an engineer.
Review of Algebra
The basics of algebra and the grammar of number – The laws of indices, the distributive law, basic algebraic operations, transposition of equations. The concept of functions and the common notation for the argument of a function. The basic functions: the logarithm and the exponential functions, the sinusoidal function and its parameters.
Review of Co-ordinate Geometry
The x-y plane. The behaviour of the common functions and their illustration by graphs in the plane. The use of the common functions to model the behaviour of physical systems such as the discharge of a capacitor, the voltage and current from a dynamo, waves and other examples. The idea of a vector; Cartesian and polar coordinates.
Review of Equations
The idea of an equation and the solutions of an equation. The solution of equations as geometric problems in the x-y plane. Simultaneous equations, Quadratic equations and Cubic equations. Complex numbers in polar and Cartesian forms.
The Derivative
The concept of the Derivative as a rate-of-change, as the slope of a curve. Establishing the derivatives of simple functions, such as linear and quadratic functions using first principles. The derivatives of basic functions encountered in the sciences and engineering, including polynomials, the logarithm and exponential functions and the basic trigonometric functions. The product and quotient rules and their application to finding derivatives of common functions found in engineering formed from the basic functions. Composite functions and the Chain rule. Maxima and Minima problems and other applications of basic calculus to geometric problems.
Integration
Integration as the inverse operation to the derivative. Identifying the integrals of common functions from this concept. The definite integral and its interpretation as the area below a curve. Applications such as the mean square and root mean square of functions in electrical science or the average power of an electrical circuit. Integration by Parts and integration by substitution, for common functions. The definition of piecewise continuous functions and the calculation of their indefinite and definite integrals.
Ordinary Differential Equations
The Methods of solution of Ordinary Differential Equations; first order linear equations and the integrating factor, separable first order equations, second order differential equations with constant coefficients and the characteristic equation. The concept of an engineering system and its modelling as a differential equation. The treatment of linear Ordinary Differential equations as systems solvable by the use of Phasors. The Laplace transform and its use in solving ordinary differential equations.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Formal Examination | 50 |
| Other Assessment(s) | 50 |