Introduction#
This is the second in a series of three short online maths books that outline useful background content for a university science degree. The material overlaps largely with the UK A-level mathematics syllabus, though with selective coverage of topics that is geared mostly towards mathematical modelling.
Book 1#
This book looks at polynomials and trigonometric functions. Both are ubiquitous in science and mathematics, since they can be used to make approximations to more complicated functions. Polynomial approximations are especially useful where we are interested in the behaviour of a function over a narrow range of values, whilst trigonometric functions are preferred where a phenomenon exhibits some inherent periodicity.
This book also includes a chapter on rational fractions, which are ratios of two polynomials. Investigating these can allow us to make comparisons between different phenomena that are each modelled using polynomials. Understanding rational fractions is also necessary background for the study of limits and differentiation.
The work in this book often draws on graphical arguments, which can help to identify and illuminate key features such as roots, turning points, and trends.
Book 2#
This book looks at exponentials and logarithms. In the book these are introduced from a data analysis perspective by thinking about concepts of growth and rates of change. We distinguish between absolute and relative growth, and outline the key exponential property of constant relative growth rate.
Crucially, this book sets out to demonstrate why the special number \(e\) is the “natural” base for exponential growth, and how growth in this base compares to growth in other bases such as doubling growth.
The unique growth properties of the exponential function make it essential for mathematical modelling of real-world phenomena. Investigating the growth properties of the exponential function also paves the road to calculus.
Book 3#
This book is your introduction to calculus, which underpins much of modern science and engineering. Calculus allows us to study the effects of change, discover relationships between variables, make predictions, and find optimal solutions to complex problems. Applications are limited only by imagination, but examples include: modelling the spread of disease, patterns in the stock market, development of populations, the relationships between reaction rates and chemical concentrations, behaviour of fluids, solids and electromagnetic or quantum fields.
The book starts with an informal overview of the concept of a limit, building on the discrete ideas that were introduced in book 2. This will allow us to arrive at a proper understanding of infinitesimal ratios, which may look like “0”/”0”. We then go on to introduce the basic ideas of differentiation and integration.