Enter the current approximation and the values of the function and its derivative at the current approximation into the calculator to determine the next approximation.

## Newton-Raphson Method Formula

The following formula is used to calculate the next approximation in the Newton-Raphson method.

x_(n+1) = x_n - f(x_n) / f'(x_n)

Variables:

• x_(n+1) is the next approximation x_n is the current approximation f(x_n) is the value of the function at x_n f'(x_n) is the value of the derivative of the function at x_n

To calculate the next approximation, evaluate the function and its derivative at the current approximation. Subtract the quotient of the function value and the derivative value from the current approximation. This gives the next approximation.

## What is a Newton-Raphson Method?

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.

## How to Calculate Newton-Raphson Method?

The following steps outline how to calculate the Newton-Raphson Method.

1. First, determine the current approximation, x_n.
2. Next, calculate the value of the function at x_n, f(x_n).
3. Next, calculate the value of the derivative of the function at x_n, f'(x_n).
4. Next, use the formula x_(n+1) = x_n – f(x_n) / f'(x_n) to calculate the next approximation, x_(n+1).
5. Repeat steps 2-4 until the desired level of accuracy is achieved.

Example Problem:

Use the following variables as an example problem to test your knowledge.

Current approximation, x_n = 2

Value of the function at x_n, f(x_n) = 5

Value of the derivative of the function at x_n, f'(x_n) = 3