Use Newton's method (see Example 8 ) to approximate a root...

Use Newton's method (see Example 8 ) to approximate a root for the given function with the specified value of $x_{0}$. Terminate your sequence when $\left|x_{n+1}-x_{n}\right|<0.001$.
$$
f(x)=x^{3}-2, x_{0}=1
$$

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Key Concepts

Newton's Method

Newton's Method is an iterative numerical technique for finding approximations to the roots of a real-valued function. It works by taking an initial guess and then iteratively improving that guess using the formula x(n+1) = x(n) - f(x(n)) / f'(x(n)), where f'(x(n)) is the derivative of the function at x(n). This method leverages the linear approximation of the function by its tangent line to quickly converge to a root.

Derivative Computation

Accurate computation of the derivative of the function is essential in Newton's Method because the derivative gives the slope of the tangent line used to approximate the function. The value of the derivative at each iteration determines the rate and direction of convergence, and any errors in its computation can cause the method to fail or converge slowly.

Iteration and Convergence Criterion

Newton's Method involves repeatedly applying the iterative formula until the successive approximations are sufficiently close to each other, indicating convergence to a root. The convergence criterion, such as requiring that the absolute difference between successive approximations is less than a given threshold, ensures that the method stops when the approximation is accurate enough.

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