### Introduction

This Section is concerned with the problem of “root location”; i.e. finding those values of $x$ which satisfy an equation of the form $f\left(x\right)=0$ . An initial estimate of the root is found (for example by drawing a graph of the function). This estimate is then improved using a technique known as the Newton-Raphson method, which is based upon a knowledge of the tangent to the curve near the root. It is an “iterative” method in that it can be used repeatedly to continually improve the accuracy of the root.

#### Prerequisites

- be able to differentiate simple functions
- be able to sketch graphs

#### Learning Outcomes

- distinguish between simple and multiple roots
- estimate the root of an equation by drawing a graph
- employ the Newton-Raphson method to improve the accuracy of a root

#### Contents

1 The Newton-Raphson method2 Finding roots of the equation $f\left(x\right)=0$

3 Engineering Example 5

3.1 Buckling of a strut