Question Bank • QB_EM_116 NAT • 2 Marks

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Solutions of linear and non-linear algebraic equations

Question Statement:

Newton-Raphson method is used to find a root of $f(x) = x^2 - 5 = 0$. If the initial guess is $x_0 = 2.0$, calculate the approximation $x_1$ after the first iteration.

Answer Key:

2.25

Step-by-Step Solution & Explanation

Newton-Raphson formula: $x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}$ Here $f(x) = x^2 - 5$, $f'(x) = 2x$. At $x_0 = 2.0$: $f(2.0) = 2^2 - 5 = -1$ $f'(2.0) = 2(2) = 4$ $x_1 = 2.0 - \frac{-1}{4} = 2.0 + 0.25 = 2.25$
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