The formula to calculate the next approximation in the Newton-Raphson method is:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
Where:
The Newton-Raphson method is a root-finding algorithm that uses iteration and linear approximation to quickly find the roots, or zeroes, of a real-valued function. It starts with an initial guess and then uses the derivative of the function to find a better estimate of the root. This process is repeated until a sufficiently accurate value is reached. The method is named after Isaac Newton and Joseph Raphson, who independently developed it in the 17th century.
Let's assume the following values:
Step 1: Evaluate the function and its derivative at the current approximation:
\[ x_{n+1} = 2 - \frac{4}{3} \approx 0.67 \]
The Next Approximation (xn+1) is approximately 0.67.
