Use Newton's method to approximate the root of $f ( x ) = x ^ { 3 } + 4 x + 2$ that lies between $x = - 1$ and $x = 0$ .
A)0.473
B)0.372
C)0.563
D)1
E)-0.473
F)-0.372
G)-0.563
H)None of these

Question

Quizplus · Accepted Answer

Newton's method uses the formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ to iteratively approximate the roots of a function. Given $f(x) = x^3 + 4x + 2$, its derivative is $f'(x) = 3x^2 + 4$. Starting with an initial guess $x_0$ between -1 and 0 (let's choose -0.5 for simplicity), we apply the formula:1. Calculate $f(-0.5) = (-0.5)^3 + 4(-0.5) + 2 = -0.125 - 2 + 2 = -0.125$.2. Calculate $f'(-0.5) = 3(-0.5)^2 + 4 = 3(0.25) + 4 = 4.75$.3. Apply Newton's method: $x_1 = -0.5 - \frac{-0.125}{4.75} \approx -0.47368$.Thus, the approximation of the root that lies between -1 and 0 is approximately -0.473, which corresponds to choice E.

Use Newton's Method to Approximate the Root Of f(x)=x3+4x+2f ( x ) = x ^ { 3 } + 4 x + 2f(x)=x3+4x+2

Use Newton's Method to Approximate the Root Of $f ( x ) = x ^ { 3 } + 4 x + 2$