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Introduction to functionsA function is a rule which operates on one number to give another number.
However, not every rule describes a valid function. This unit explains
how to see whether a given rule describes a valid function, and introduces
some of the mathematical terms associated with functions.
Exponential and logarithm functionsExponential functions and logarithm functions are important in both theory and
practice. In this unit we look at the graphs of exponential and logarithm
functions, and see how they are related.
Trigonometric equationsThis workbook explains how a number of trigonometric equations can be solved by making reference to a Table of standard results and using the symmetries and periodicities present in the graphs of trig functions.
Integration by partsA special rule, integration by parts, is available for integrating
products of two functions. This unit derives and illustrates this rule with a
number of examples.
Completing the squareIt is often useful to be able write a quadratic expression in an alternative form - that is as a complete square plus or minus a number. The process for doing this is called completing the square. This booklet explains how this process is carried out.
Partial fractionsAn algebraic fraction can often be broken down into the sum of simpler fractions called partial fractions. This process is required in the solution of a number of engineering and scientific problems.
This booklet explains how this is done.
Polar co-ordinatesThe (x, y) co-ordinates of a point in the plane are called its Cartesian
co-ordinates. But there is another way to specify the position of a point, and
that is to use polar co-ordinates (r, theta). In this unit we explain how to
convert from Cartesian co-ordinates to polar co-ordinates, and back again.
Limits of functionsIn this unit, we explain what it means for a function to tend to infinity,
to minus infinity, or to a real limit, as x tends to infinity or to minus
infinity. We also explain what it means for a function to tend to a real limit
as x tends to a given real number. In each case, we give an example of a
function that does not tend to a limit at all.
DecimalsIn this unit we shall look at the meaning of decimals, and how they are related
to fractions. We shall then look at rounding to given numbers of decimal places
or significant figures. Finally we shall take a brief look at irrational numbers.
Fractions - basic ideasWhat are fractions? Fractions are ways of writing parts of whole numbers. For
example if we take a pizza, and divide it up equally between 4 people, each
person will have 1/4 or, written in words, one quarter of the pizza.
Polynomial divisionPolynomial division is a process used to simplify certain sorts of algebraic fraction. It is very similar to long division of numbers. This booklet describes how the process is carried out.
RatiosA ratio is a way of comparing two or more similar quantities, by writing two
or more numbers separated by colons. The numbers should be whole numbers, and
should not include units.
Rules of arithmeticEvaluating expressions involving numbers is one of the basic tasks in
arithmetic. But if an expression is complicated then it may not be clear
which part of it should be evaluated first, and so some rules must be
established. There are also rules for calculating with negative numbers.
Surds and other rootsRoots and powers are closely related, but only some roots can be written as
whole numbers. Surds are roots which cannot be written in this way.
Nevertheless, it is possible to manipulate surds, and to simplify formulae.
