Question

Let $f(x) = x^3 + 2x^2 + 3x - 1$. Newton-Raphson method is used to find the root of the equation $f(x) = 0$. If the initial guess is $x_0 = 1$, then the value of $x$ after first iteration is ...........

• A. 0.30
• B. 0.36
• C. 0.40
• D. 0.50

Hint:

In the Newton-Raphson method, remember to calculate both $f(x)$ and $f'(x)$ at each iteration to refine the estimate of the root.

Answer 1

The Correct Option is D

The Newton-Raphson method is given by the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] First, calculate the first derivative of $f(x)$: \[ f'(x) = 3x^2 + 4x + 3 \] Now, use the initial guess $x_0 = 1$: \[ f(1) = 1^3 + 2(1)^2 + 3(1) - 1 = 1 + 2 + 3 - 1 = 5 \] \[ f'(1) = 3(1)^2 + 4(1) + 3 = 3 + 4 + 3 = 10 \] Now, apply the Newton-Raphson formula: \[ x_1 = 1 - \frac{5}{10} = 1 - 0.5 = 0.50 \] Thus, the value of $x$ after the first iteration is $0.50$.