1 The Newton-Raphson method

We first remind the reader of some basic notation: If $f\left(x\right)$ is a given function the value of $x$ for which $f\left(x\right)=0$ is called a root of the equation or zero of the function. We also distinguish between various types of roots: simple roots and multiple roots. Figures 21 - 23 illustrate some common examples.

Figure 21 Figure 22 Figure 23

More precisely; a root $x_0$ is said to be:

a simple root if $f\left(x_0\right)=0\text{and}\frac{df}{dx}{\left|\right.}_{x_0}\ne 0.$

a double root if $f\left(x_0\right)=0,\frac{df}{dx}{\left|\right.}_{x_0}=0\text{and}\frac{d^{2}f}{d{x}^2}{\left|\right.}_{x_0}\ne 0,$ and so on.

In this Section we shall concentrate on the location of simple roots of a given function $f\left(x\right)$ .

Task!

Given graphs of the functions

1. $f\left(x\right)=x^3-3x^2+4$ ,
2. $f\left(x\right)=1+sinx$ classify the roots into simple or multiple.

Answer

Answer