Question Bank • QB_EM_ADV_011 NAT • 2 Marks

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Solutions of linear and non-linear algebraic equations

Question Statement:

The root of the equation $f(x) = x^3 - 2x - 5 = 0$ is computed using the Newton-Raphson method starting with an initial guess $x_0 = 2$. The value of $x_2$ after two iterations is ________ (round off to three decimal places).

Answer Key:

2.095

Step-by-Step Solution & Explanation

Derivative: $f'(x) = 3x^2 - 2$. Newton-Raphson iteration formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ Iteration 1 ($x_0 = 2$): $f(2) = 2^3 - 2(2) - 5 = 8 - 4 - 5 = -1$ $f'(2) = 3(2^2) - 2 = 12 - 2 = 10$ $x_1 = 2 - \frac{-1}{10} = 2.100$ Iteration 2 ($x_1 = 2.1$): $f(2.1) = (2.1)^3 - 2(2.1) - 5 = 9.261 - 4.2 - 5 = 0.061$ $f'(2.1) = 3(2.1)^2 - 2 = 3(4.41) - 2 = 13.23 - 2 = 11.23$ $x_2 = 2.1 - \frac{0.061}{11.23} = 2.1 - 0.005432 = 2.094568 \approx 2.095$
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