Maths for Engineering: Notes for School Teachers
The choice of a Mathematics A Level course to follow, and the choice of options or modules within that course, is an important decision for any school. That choice will be influenced by many factors, one of which is suitability for the needs of those students intending to go on to university courses in science subjects. We frequently receive enquiries from schools, either directly or via College Admissions Offices, as to the particular requirements for those entering our undergraduate course in Engineering. The assumed knowledge at the start of our first-year mathematics course is set out below in the form of a syllabus (which is based very closely on the national Core Syllabus in Mathematics, which was proposed early in the 1990s).
It has been our practice for many years to distribute to all incoming students a booklet of practice problems (drawn from past A Level papers), together with detailed book references for self-study. Those who have missed some topics are strongly advised to use this booklet to make good the deficit. However, it is of obvious benefit to them if they have been given a thorough grounding in the elements of these topics as part of their school course.
Laws of indices (including negative and rational exponents); addition, subtraction, multiplication and factorisation of polynomials; the factor theorem; the use of the modulus sign; solution of linear and quadratic inequalities and equations; solution of simultaneous linear equations.
Cartesian coordinates in two and three dimensions; distance between points; Cartesian basis vectors i, j, k and representation by vectors; equation of a straight line in the form y = mx + c; finding an equation of a linear graph; Cartesian and parametric equations of curves.
The idea of complex numbers; real and imaginary parts, modulus and argument; addition, subtraction, multiplication and division of complex numbers; the Argand diagram.
Representation of two simultaneous linear equations with a 2x2 matrix; addition and multiplication of 2x2 matrices; determinant and inverse of 2x2 matrices.
Graphical representations and their inverses; understanding domain, range and composition; knowledge of the effect of simple scaling and shifting transformations of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax).
Sequences and series
Definition of sequences; recognition of periodicity, oscillation, convergence and divergence; arithmetic and geometric series; sum to infinity of a convergent geometric series; binomial expansion of (1 + x)n.
Radian measure; the formulae s = rt and A = r2t/2, for length r and angle t. Sine, cosine, and tangent functions, their graphs, symmetries and periodicity; formulae for sin(A + B), sin(A - B), cos(A + B), and cos(A - B); double angle formulae.
Exponentials and logarithms
Definition and properties of ex and ln x including their graphs; laws of logarithms; exponential growth and decay; the solution of equations of the form ax = b.
The concept of derivative as gradient; differentiation and its relation to the idea of a limit; algebraic differentiation of polynomials; differentiation of ex, ln x, sin x, cos x and tan x; application of differentiation to gradients, maxima and minima and stationary points; increasing and decreasing functions; rates of change; differentiation of functions using sums, differences, products, quotients and composition; differentiation of inverse functions; formation of simple differential equations.
The concept of integration: evaluation of area under a curve; integration as the inverse of differentiation; integration of xn, ex, 1/x, sin x, cos x; evaluation of definite integrals with fixed limits; simple cases of integration by substitution and by parts; solution of simple differential equations by either analytic or numerical means.
The application of mathematics
Understanding the process of mathematical modelling with reference to one or more application areas; abstraction from a real-world situation to a mathematical description; the selection and use of a simple mathematical model to describe a real-world phenomenon; approximation, simplification and solution; interpretation and communication of mathematical results and their implications in real-world terms; progressive refinement of mathematical models.