Question

Use Newton’s Method to find the solution to the equation

-x^2+18=0 use x_1=3 and find x_4 accurate to six decimal places.

Answer 1

`f(x) = - x {}^(2) + 18 \\ f'(x) = - 2x`

`x_(n + 1) = x_(n) - (f(x_(n)))/(f'(x_(n)))`

`x_(2) = 3 - (f(3))/(f'(3)) = 4.5`

`x _(3)= 4.5 - (f(4.5))/(f'(4.5)) = 4.25`

`x_(4) = 4.25 - (f(4.25))/(f'(4.25)) ≈4.242647`

`x_(5) = 4.242647 - (f(4.242647))/(f'(4.242647)) ≈4.242640 \\`