Question

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

Answer 1

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

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

$x_{2} = 3 - \frac{f(3)}{f'(3)} = 4.5$

$x _{3}= 4.5 - \frac{f(4.5)}{f'(4.5)} = 4.25$

$x_{4} = 4.25 - \frac{f(4.25)}{f'(4.25)} ≈4.242647$

$x_{5} = 4.242647 - \frac{f(4.242647)}{f'(4.242647)} ≈4.242640$