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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The correct answer is 1.909.

Key Points

  • The Newton-Raphson method is an iterative numerical method used to find the roots of a real-valued function.
  • Given a function f(x) and its derivative f(x) , the Newton-Raphson iteration formula is:
    xn+1=xnf(xn)f(xn)
  • For the given equation f(x)=x3x5 , we first need to compute its derivative:
    f(x)=3x21
  • Starting with the initial approximation x0=2 , we apply the Newton-Raphson iteration formula:
    x1=x0f(x0)f(x0)
  • Substituting x0=2 :
    f(2)=2325=825=1
    f(2)=3(22)1=3(4)1=121=11
    x1=2111=20.0909=1.909

Additional Information

  • The Newton-Raphson method is highly efficient and converges quickly if the initial approximation is close to the actual root.
  • It is important to ensure that the derivative f(x) is not zero at the initial approximation to avoid division by zero.
  • The method may fail to converge or converge to a different root if the initial approximation is not chosen carefully.
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