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] : 

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: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$
• For the given equation $f(x) = x^3 - x - 5$ , we first need to compute its derivative: $$f'(x) = 3x^2 - 1$$
• Starting with the initial approximation $x_0 = 2$ , we apply the Newton-Raphson iteration formula: $$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$$
• Substituting $x_0 = 2$ : $$f(2) = 2^3 - 2 - 5 = 8 - 2 - 5 = 1$$ $$f'(2) = 3(2^2) - 1 = 3(4) - 1 = 12 - 1 = 11$$ $$x_1 = 2 - \frac{1}{11} = 2 - 0.0909 = 1.909$$

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.