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Version 1. Changes include the amendment of section 5. Please note that these are not side-barred. GCE Mathematics Further Pure unit 3 MFP3. The Assessment and Qualifications Alliance AQA is a company limited by guarantee registered in England and Wales and a registered charity number Dr Michael Cresswell Director General. All rights reserved. Further Pure 3: Contents. Chapter 1: Series and limits.

Chapter 2: Polar coordinates. Chapter 3: Introduction to differential equations. Chapter 4: Numerical methods for the solution of first order differential equations. Chapter 5: Second order differential equations. Differential equations of the form. Answers to the exercises in Further Pure 3. Further Pure 3: Introduction. The aim of this text is to provide a so und and readily accessible account of the items comprising the Further Pu re Mathematics unit 3. The chapters are arranged in the same order as the five main sections of the unit.

The first chapter is therefore concerned w ith series expansions and the evaluation of limits and improper integrals. The second covers polar coordinates and their use in curve sketching and evaluation of areas. The subject of differential equati ons forms a major part of this unit and Chapters 3, 4 and 5 are devoted to this topic.

Chapter 3 introduces the subject and deals mainly with analytical methods for solving differential equations of first order linear form.

In addition to the standard method of solution using an integrating factor , this chapter introduces the method based on finding a complementary function and a particular integral. This provide s useful preparation for Chapter 5 where the same technique is used for solving second order differential equations.

With the advent of modern computers, numerical methods have become an essential practical tool for solving the many differential equations wh ich cannot be solved by analytical methods. This important subject is covered in Chapter 4 in relation to differential equations of the form. It should be appreciated that, in prac tice, the numerical methods described would.

The purpose of the worked examples and exercises in this text is to exemplify the principles of the various methods and to show how these methods work. Relatively simple functions have been chosen, as far as possible, so that th e necessary calculations with a sc ientific calculator are not unduly tedious. Chapter 5 deals with analytical methods for solv ing second order differential equations and this requires some knowledge of complex numbers.

Pa rt of the required knowledge is included in the Further Pure 1 module, which is a prerequisite for studying this module, and the remainder is included in the Further Pure 2 module which is not a prerequisite. For both simplicity and completeness therefore, Chapter 5 begins with three short sections on complex numbers which cover, in a straightforward way, all that is required for the purpos e of this chapter. These sections should not cause any difficulty and it is hoped that they will be found interesting as well as useful.

Those who have already studied the topics covered can either pass over this work or regard it as useful revision.

The main methods for solving second or der linear differential equations with constant coefficients are covered in Sections 5.

These methods sometimes seem difficult when first met, but students should not be discouraged by this. Useful summaries are highlighted in the text and confidence should be restored by studying how these are applied in the worked examples and by working through the exercises. The text concludes with a short section show ing how some second or der linear differential equations with variable coefficients can be solved by using a substitution to transform them to simpler forms.

Chapter 1: Series and Limits. In this chapter, it is shown how series expa nsions are used to find limits and how improper integrals are evaluated. When you have completed it you will:.

You will have already met the idea of a limit and be familiar with some of the notation used. This may be expressed as. The two cases are illustrated in the diagram below.

The distinction between the two cases is impor tant, for instance, when we consider the. In some simple cases, it is easy to see how a function f x behaves as x approaches a given value and whether it has a limit. Here are three examples. The first two of the above examples can be expressed as:. However, it would be wrong to state that. The function 1. Another example, not quite so straightforward as the examples above, is that of finding the.

It can now be seen that f. The limiting value of f x is. The working for this example can be presented more concisely as follows. There are many instances where the behaviour of a function is much more difficult to determine than in the cases consider ed above. For example, consider. It is not obvious therefore what happens. Consider also the function. Investigating with a calculat or will produce the following results, to five decimal places. However, no matter how convincin g the evidence may seem, a numerical investigation of this kind does not constitute a satisfactory mathematical proof.

Limits in these more difficult cases can often be found with the help of series expansions. The series expansions that we shall use are introduced in the next thr ee sections of this chapter. Exercise 1A. Write down the values of the following limits.

Find the values of the following. Use a calculator to investigate the behaviour of x. You will find it. Later in this chapter, it will be shown that the function tends to zero. You will already be famili ar with the binomial series expansion for 1. Such expansions are called Maclaurin series. The Maclaurin series for a function f x is given by:.

To derive this, the follow ing assumptions are made. The function f x can be expressed as a series of the form. The series can be differentiated term by term. Succesive differentiations of each side of the equation under i gives. Substituting these values into the series in i above gives the Maclaurin series of f x. The Maclaurin series has an interesting history. It is named in honour of Colin Maclaurin, a notable Scottish mathematician. Born in , he was a child prodigy who entered university at the age of 11 and became a pr ofessor at the age of Example 1.

In this case. Substituting these values into the genera l form of the Maclaurin series gives. The general term of this series is most readily obtained by inspecting the first few terms. It is. Obtain the Maclaurin series expansion of sin x. By inspecting the first few terms of the series, the general term can be identified.

The general term could also be expressed as. Whenever the general term of a se ries is given, the admissible values of r should be stated. Exercise 1B. Obtain the Maclaurin series expansion of e x , up to and including the term in.

Show that the. Write down an expression for the general term of the series. Give the general term. By using a similar method to th at used to derive the Maclaurin series of f x , show that. It is named after Brook Taylor, an eminent English mathematician who was a clos e contemporary of Maclaurin. The Maclaurin series expa nsion of a function f x is not necessarily va lid for all values of x.

A simple example will show this. To determine the values of x for which the expansion is valid is not too difficult in this case.

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