GATE AE 2007 • Q48 MCQ • 2 Marks Engineering Mathematics

Engineering Mathematics

Numerical Methods

Subtopic: Newton-Raphson Method

Question Statement:

The Newton-Raphson iteration formula to find the cube root of a positive number $c$ ($x = c^{1/3}$) is:

Options:

  • (A) $x_{k+1} = \frac{2x_k^3 + c}{3x_k^2}$
  • (B) $x_{k+1} = \frac{2x_k^3 - c}{3x_k^2}$
  • (C) $x_{k+1} = \frac{x_k^3 + 2c}{3x_k^2}$
  • (D) $x_{k+1} = \frac{2x_k + c}{3x_k^2}$

Official Answer Key:

A

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2007 Q.48)** **1. Function Definition**: We seek the root of: $$f(x) = x^3 - c = 0$$ The derivative is: $$f'(x) = 3x^2$$ **2. Newton-Raphson Iteration**: $$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)} = x_k - \frac{x_k^3 - c}{3x_k^2} = \frac{3x_k^3 - (x_k^3 - c)}{3x_k^2} = \frac{2x_k^3 + c}{3x_k^2}$$ **3. Official Master Key**: - **Official Answer Key**: **A** ($x_{k+1} = \frac{2x_k^3 + c}{3x_k^2}$)
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