Further Pure 3

 

Chapter 1 - Series and Limits (18.1)

1.1 The concept of a limit

1.2 Finding limits in simple cases

1.3 Maclaurin’s series expansion

1.4 Range of validity of a series expansion

1.5 The basic series expansions

1.6 Use of series expansions to find limits

1.7 Two important limits

1.8 Improper integrals

Use of the range of values of x for which these expansions are valid, as given in the formulae booklet, is expected to determine the range of values for which expansions of related functions are valid;

 

In this chapter, it is shown how series expansions are used to find limits and how improper integrals are evaluated.

When you have completed it you will:

  • have been reminded of the concept of a limit;

  • have been reminded of methods for finding limits in simple cases;

  • know about Maclaurin’s series expansion;

  • be able to use series expansions to find certain limits;

  • know about the limits of xkex as x→∞ and xkln x as x→0 ;

  • know the definition of an improper integral;

  • know how to evaluate improper integrals by finding a limit.

Chapter 2 - Polar Relationships (18.2)

2.1 Cartesian and polar frames of reference

2.2 Restrictions on the values of θ

2.3 The relationship between Cartesian and polar coordinates

2.4 Representation of curves in polar form

2.5 Curve sketching

2.6 The area bounded by a polar curve

This chapter introduces polar coordinates.

When you have completed it, you will:

  • know what is meant by polar coordinates;

  • know how polar coordinates are related to Cartesian coordinates;

  • know that equations of curves can be expressed in terms of polar coordinates;

  • be able to sketch curves of equations given in polar form;

  • be able to find areas by integration using polar coordinates.

 

Chapter 3 - Differential Equations (18.3)

3.1 The concept of a differential equation: order and linearity

3.2 Families of solutions, general solutions and particular solutions

3.3 Analytical solution of first order linear differential equations: integrating factors

3.4 Complementary functions and particular integrals

3.5 Transformations of non-linear differential equations to linear form

The relationship of order to the number of arbitrary constants in the general solution will be expected.

This is the first of three chapters on differential equations.

When you have completed it you will:

  • have been reminded of the basic concept of a differential equation;

  • have been reminded of the method of separation of variables;

  • have been reminded of the growth and decay equations;

  • understand the terms order, linearity, families of solutions, general solutions, particular solutions, boundary conditions, end conditions and initial conditions;

  • know how to solve first order linear differential equations using an integrating factor;

  • know how to solve first order linear differential equations with constant coefficients by finding a complementary function and a particular integral;

  • know how some first order non-linear differential equations can be solved by transforming them to linear form.

Chapter 4 - Differential Equations – 1st Order (18.4) - Khan Academy Videos

4.1 Introduction

4.2 Euler’s formula

4.3 The mid-point formula

4.4 The improved Euler formula

4.5 Error analysis: some practical considerations

To include use of an integrating factor and solution by complementary function and particular integral.

Formulae to be used will be stated explicitly in questions, but candidates should be familiar with standard notation such as used in:

This chapter gives an introduction to numerical methods for solving first order differential equations.

When you have completed it, you will:

  • be familiar with the standard notation used;

  • be familiar with the methods which use Euler’s formula, the mid-point formula and Euler’s improved formula;

  • know how the above formulae can be derived both geometrically and analytically;

  • be aware of the principal sources of error in the methods described;

  • know how the accuracy of a numerical solution can be estimated.

Chapter 5 - Differential Equations – 2nd Order (18.5) - Khan Academy Videos

5.1 Introduction to complex numbers

5.2 Working with complex numbers

5.3 Euler’s identity

5.4 Formation of second order differential equations

5.5 Differential equations of the form

5.6 Differential equations of the form

5.7 Solution of second order differential equations by reduction to simultaneous first order differential equations

Including repeated roots.

Finding particular integrals will be restricted to cases where f (x) is of the form ekx , cos kx , sin kx or a polynomial of degree at most 4, or a linear combination of any of the above.

Level of difficulty as indicated by:-

(a) Given x2d2y/dx2 – 2y = x use the substitution x = et

to show that d2y/dt2 – dy/dt – 2y = et

Hence find y in terms of t

Hence find y in terms of x

(b) Use the substitution u = dy/dx to show that

and hence that where A is an arbitrary constant.

Hence find y in terms of x.

This chapter deals with analytical methods for solving second order linear differential equations with constant coefficients.

When you have completed it, you will:

  • have been introduced to the concept of complex numbers, which prove useful in the analytical methods described;

  • know sufficient about complex numbers, including Euler’s identity, for the purposes of this chapter;

  • be able to solve differential equations of the form using the auxiliary equation ak2 + bk + c = 0;

  • be able to solve differential equations of the form by finding a complementary function and a particular integral;

  • know how second order linear differential equations can be solved by reduction to simultaneous first order linear differential equations.