Question

find the roots using newton raphson. Stop until at 15th iteration or when approximate percent relative error is below 0.05%.

e^x+x=4

Answer 1

$$f(x)=e^x+x-4$$

$$f'(x)=e^x+1$$

$$x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}$$

Initial solution $x_0 =1$

$$\begin{matrix} n & x_n & f(x_n) & x_{n+1} \\ 0 & 1 & -0.281718 & 1.075766\\ 1 & 1.075766 & 0.008003 & 1.073729 \\ 2 & 1.073729 & 0.000006 & 1.073729 \\ \end{matrix}$$

$$\varepsilon =\big|\dfrac{x_{n+1}-x_n}{x_n}\big|\cdot 100\%$$

$$\varepsilon =\big|\dfrac{1.075766-1}{1}\big|\cdot 100\%=7.5766\%$$

$$\varepsilon =\big|\dfrac{1.073729-1.075766}{1.075766}\big|\cdot 100\%=0.1894\%$$

$$\varepsilon =\big|\dfrac{1.073729-1.073729}{1.073729}\big|\cdot 100\%=0.000000\%$$

$x=1.073729$

The root is $1.073729$