Question

The formula to solve simple nonlinear equation \(f(x) = 0\) by Newton's Raphson Method is given by

• A. \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)
• B. \(x_{n+1} = x_n - \frac{f'(x_n)}{f(x_n)}\)
• C. \(x_{n+1} = f(x_n) - \frac{f'(x_n)}{f'(x_n)}\)
• D. \(x_{n+1} = f(x_n) - \frac{f'(x_n)}{f'(x_n)}\)

Hint:

Remember the formula: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).
Ensure \(f'(x_n) \neq 0\) to avoid division by zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Newton's Raphson method is an iterative numerical technique
for finding roots of real-valued functions.

Step 2: Key Formula or Approach:
The formula is derived from Taylor's series expansion.

Step 3: Detailed Explanation:
The Newton-Raphson formula is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. \]
This formula is used to approximate the root of \(f(x) = 0\).
Option (A) is correct.
Option (B) has the numerator and denominator swapped.
Options (C) and (D) are incorrect as they do not follow the standard form.
The method requires an initial guess \(x_0\).
It converges quadratically if the initial guess is close to the root.
The derivative \(f'(x_n)\) must be non-zero.
This method is widely used for solving equations in engineering and science.