Which mathematical method can be used to approximate real zeros of a continuous function?

Newton's method is a powerful numerical technique used to approximate real zeros of a continuous function. This method involves taking an initial guess for the root of the function and then iteratively improving that guess using the function's derivative. The formula used in Newton's method is ( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} ), where ( x_n ) is the current approximation, ( f(x_n) ) is the value of the function at that point, and ( f'(x_n) ) is the value of the derivative.

This method is particularly efficient because it can converge rapidly to a solution, often requiring only a few iterations if the initial guess is close to the actual root. However, its effectiveness can depend on the initial choice of starting point, especially in cases where the function has steep slopes or points of inflection.

While other methods like graphing or the bisection method also serve to approximate zeros, they do not offer the same level of precision and speed as Newton's method when conditions are favorable. Graphing provides a visual approach to identify where the function crosses the x-axis but is limited by the resolution of the graph, and the bisection