How to find the cube root of 10 using Newton Raphson Method

Question

How to find the cube root of $10$ by Newton-Raphson method correct up to $5$ significant figures?

Answer 1

Start by defining $f(x) = x^3 - 10$. By simple differentiation, $f'(x)=3x^2$.

You want to find the single real zero of $f(x)$.

You can start with an obvious initial estimate of $x_0=2$.

Now apply Newton-Raphson, which iteratively gives $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$.

Do this until the last significant digit required stops changing.