233333 Let x0 2and use Newtons Method to calculate the next

2.33333 Let x_0 = 2|and use Newton\'s Method to calculate the next four approximations to a squareroot of x^4 - 6x^2 +x + 3.| (Use decimal notation. Give your answers to five decimal places.)

Solution

Acording to Newtons method;

Xn+1 = Xn - f(Xn) / f\'(Xn)

Given Equation is

F(X) = X^{4 _}6X² + X + 3

F\'(X) = 4X³ - 12X + 1

X₀ = 2

X₁ = X₀ - f(X₀) / f\'(X_o)

X1 = 2 - (-3) / 9

X1 = 2.33333

X2 = X1 - f(X1) / f\'(X1)

= 2.33333 - (2.30856/23.81463)

X2 = 2.23639

X3 = X2 - f(X2) / f\'(X2)

= 2.23639 - (0.24215/18.90400)

X3 = 2.22358

X4 = X3 - f(X3)/f\'(X3)

= 2.22358 - (3.9136* 10pow-3)/ 18.29329

X4 = 2.22336