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The real root of the equation x3 − x − 5 = 0 lying between 1 and 2 after first iteration by Newton - Raphson method is _______, if initial approximation is taken as x0 = 2∈ [1, 2] : 

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  1. 1.909 
  2. 1.904 
  3. 1.921 
  4. 1.940

Answer (Detailed Solution Below)

Option 1 : 1.909 
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Detailed Solution

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The correct answer is option 1: 1.909

Key Points

We are given the equation:

Its derivative is:

Initial approximation:

Newton-Raphson iteration formula is:

Step-by-step Calculation:

But this gives 1.909, which matches option 1.

However, options provided show 1.904 as correct (option 2), but mathematically, the correct value after first iteration is:

Additional Information

  • The Newton-Raphson method converges quadratically near the root.
  • This method is highly efficient for equations with simple real roots.
  • Only one iteration gives an approximate root: 1.909

Hence, the correct answer is: option 1: 1.909

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