Question

Using Newton’s method to find the zero of the function f(x)=x5−3x3+1 using x0=0.7 , which of the following best represents the value of x2 ?

Answer 1

x2=0.7136 x 2 = 0 . 7136 x2=0.7273 x 2 = 0 . 7273 x2=0.7369 x 2 = 0 . 7369 x2=0.7402 x 2 = 0 . 7402 x2=0.7418

Answer 2

To use Newton's method, we start with an initial value x0 and iteratively update the value using the formula:

x1 = x0 - f(x0)/f'(x0)

We can find the derivative of the function f(x) = x^5 - 3x^3 + 1 using the power rule:
f'(x) = (5x^4) - (9x^2)

Given x0 = 0.7, we can calculate x1:
x1 = x0 - f(x0)/f'(x0)
x1 = 0.7 - (0.7^5 - 3(0.7)^3 + 1) / (5(0.7)^4 - 9(0.7)^2)

To find x2, we repeat the process using x1 as the initial value. Therefore:
x2 = x1 - f(x1)/f'(x1)

Using a calculator or computer software, we can calculate the value of x2 as follows:

x2 = x1 - f(x1)/f'(x1)
= x1 - (x1^5 - 3x1^3 + 1) / (5x1^4 - 9x1^2)

Thus, to find the value of x2, we need to evaluate the above expression using the calculated value of x1.

Answer 3

Based on the given options, the best representation of the value of x2 using Newton's method is x2 = 0.7418.