Subtopic: Solutions of linear and non-linear algebraic equations
Question Statement:
The positive root of the algebraic equation $f(x) = x^2 - 7 = 0$ is computed using the Newton–Raphson method starting with an initial guess $x_0 = 3$. The estimate $x_1$ after the first iteration is ________ (round off to three decimal places).
Answer Key:
2.667
Step-by-Step Solution & Explanation
The Newton–Raphson iteration formula is:
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
For $f(x) = x^2 - 7$, the derivative is $f'(x) = 2x$.
Starting with $x_0 = 3$:
$f(3) = 3^2 - 7 = 9 - 7 = 2$
$f'(3) = 2(3) = 6$
$x_1 = 3 - \frac{2}{6} = 3 - \frac{1}{3} = \frac{8}{3} \approx 2.6667 \approx 2.667$