Question Bank • QB_EM_NM_002 NAT • 2 Marks

Section 1: Engineering Mathematics

Numerical Methods

Subtopic: Solutions of linear and non-linear algebraic equations

Question Statement:

Using the Newton–Raphson method to find the root of $f(x) = x^2 - 5 = 0$ with an initial approximation $x_0 = 2$, the value of the next approximation $x_1$ is ________ (round off to 2 decimal places).

Answer Key:

2.25

Step-by-Step Solution & Explanation

The Newton–Raphson formula is: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ Here $f(x) = x^2 - 5$, so $f'(x) = 2x$. For $x_0 = 2$: $f(2) = 2^2 - 5 = 4 - 5 = -1$ $f'(2) = 2(2) = 4$ Substituting into the iteration formula: $x_1 = 2 - \frac{-1}{4} = 2 + 0.25 = 2.25$
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