Question

Use Newton's method to approximate a root of the equation 4 x⁷+7 x⁴+4=0 as follows. Let x_1=2 be the initial approximation. The second approximation x_2 is

Answer 1

Using Newton's method, with an initial approximation of x_1=2, the second approximation for the root of the equation 4x⁷+7x⁴+4=0 is approximately 1.67095588235294.

Step-by-step explanation:

To find the second approximation (x_2) using Newton's method for the equation 4x⁷+7x⁴+4=0, we first need the function f(x) and its derivative f'(x). In this case, f(x) = 4x⁷+7x⁴+4. The derivative, f'(x), is 28x⁶+28x⁳. Newton's method uses the formula: xn+1 = xn - f(xn)/f'(xn). Given that x_1=2, we substitute to get x_2 = 2 - f(2)/f'(2).

Calculating both f(2) and f'(2), we find:

• f(2) = 4(2)⁷+7(2)⁴+4 = 4(128)+7(16)+4 = 716,
• f'(2) = 28(2)⁶+28(2)⁳ = 28(64)+28(8) = 2176.

Subsequently, we calculate x_2:

x_2 = 2 - 716/2176 = 2 - 0.329044117647059 = 1.67095588235294.

Therefore, the second approximation x_2 is approximately 1.67095588235294.