Question

use newtons method to estimate the solutions of the equation x^2+x-1 start with x0=-1 for left and x0=1 for right hand solution find x2

Answer 1

\[x_2=x_1-\frac{f(x_1)}{f'(x_1)}\] so lets compute the numbers

Answer 2

if \[x_0=1\] then \[f(1)=1+1-1=1\] and
\[f'(x)=2x+1\] so \[f'(1)=3\]

Answer 3

therefore
\[x_1=1-\frac{1}{3}=\frac{2}{3}\]

Answer 4

to find \(x_2\) compute
\[\frac{2}{3}-\frac{f(\frac{2}{3})}{f'(\frac{2}{3})}\] i would use a calculator, but you could do it by hand

Answer 5

and then would we do the same but with -1 in order to find the left hand solution as well?