Use Newton's Method to find the largest solution of x⁵

Question

Use Newton's Method to find the largest solution of x⁵ - 5x⁴ + 4x³-x² + 2x + 10 = 0. A) 1.8235294 B) 1.8038147 C) 1.8035400 D) 1.8035520 E) 1.8035532

Quizplus · Accepted Answer

Using Newton's method, we start with an initial guess of x = 1.8:

$f(x) = xS_1 U_1 P_1 5S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1 0.5 xS_1 U_1 P_1 4S_1 S_1 P_0 + 0.4 xS_1 U_1 P_1 3S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1 xS_1 U_1 P_1 2S_1 S_1 P_0 + x + 10$

$f'(x) = S_1 U_1 P_1 5S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1 0.5S_1 U_1 P_1 4S_1 S_1 P_0 + 0.4S_1 U_1 P_1 3S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1S_1 U_1 P_1 2S_1 S_1 P_0 + 1$

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1.8 - \frac{1.8S_1 U_1 P_1 5S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1 0.5 	imes 1.8S_1 U_1 P_1 4S_1 S_1 P_0 + 0.4 	imes 1.8S_1 U_1 P_1 3S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1 1.8S_1 U_1 P_1 2S_1 S_1 P_0 + 1 + 10}{S_1 U_1 P_1 5S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1 0.5S_1 U_1 P_1 4S_1 S_1 P_0 + 0.4S_1 U_1 P_1 3S_1 S_1 P_0 F_1 F_1 F_1 S_1 - F_1 F_1 F_1S_1 U_1 P_1 2S_1 S_1 P_0 + 1}$

$x_1 \approx 1.80355201$

We then continue running Newton's method until convergence. We find that the largest solution of the equation is approximately 1.80355201, which corresponds to choice (D).

Use Newton's Method to Find the Largest Solution of X5