Question

The irrational number \(\sqrt{2}\) can be approximated by applying Newton's method to the nonlinear equation \(f(x) = x^2 - 2 = 0\). What is the Newton iteration formula?

• A. \(x_{k+1} = x_k + \frac{x_k^2 - 2}{2x_k}\)
• B. \(x_{k+1} = x_k - \frac{x_k^2 - 2}{2x_k}\)
• C. \(x_{k+1} = x_k - \frac{x_k^2 - 2}{x_k}\)
• D. \(x_{k+1} = x_k + \frac{x_k^2 - 2}{x_k}\)

Hint:

The Newton-Raphson formula always has a minus sign: \(x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}\).
This immediately rules out Options (A) and (D).
Since the derivative of \(x^2 - 2\) is \(2x\), the denominator must contain \(2x_k\), which leaves Option (B) as the only mathematically correct choice.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical formula for the Newton-Raphson iteration process when applied to solve the specific non-linear equation \(f(x) = x^2 - 2 = 0\), which is used to approximate the value of \(\sqrt{2}\).

Step 2: Key Formula or Approach:
The Newton-Raphson method is an iterative numerical technique used to find the roots of a real-valued function \(f(x) = 0\).
The general iterative formula is given by:
\[ x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)} \] where:
\(x_k\) is the current estimate of the root.
\(x_{k+1}\) is the next, improved estimate of the root.
\(f'(x_k)\) is the derivative of the function evaluated at the current estimate.

Step 3: Detailed Explanation:
Let us apply the general Newton-Raphson formula to the given function:
1. Identify the function \(f(x)\):
\[ f(x) = x^2 - 2 \] 2. Compute the derivative of the function, \(f'(x)\), with respect to \(x\):
\[ f'(x) = \frac{d}{dx}(x^2 - 2) = 2x \] 3. Express these functions in terms of the current iteration variable \(x_k\):
\[ f(x_k) = x_k^2 - 2 \] \[ f'(x_k) = 2x_k \] 4. Substitute these expressions into the generic Newton-Raphson iterative formula:
\[ x_{k+1} = x_k - \frac{x_k^2 - 2}{2x_k} \] This matches Option (B) exactly.
- Note on simplification: If we simplify this expression further, we get:
\[ x_{k+1} = x_k - \frac{x_k^2}{2x_k} + \frac{2}{2x_k} = x_k - \frac{x_k}{2} + \frac{1}{x_k} = \frac{1}{2}\left(x_k + \frac{2}{x_k}\right) \] This is the classic Babylonian method for square roots, but the options present the unsimplified form, which corresponds directly to Option (B).

Step 4: Final Answer
The Newton iteration formula is \(x_{k+1} = x_k - \frac{x_k^2 - 2}{2x_k}\), which corresponds to option (B).