Complete two iterations of Newton's Method for the function using the initial guess . Round all numerical values in your answer to four decimal places.

Question

Complete two iterations of Newton's Method for the function  using the initial guess  . Round all numerical values in your answer to four decimal places.

Quizplus · Accepted Answer

Using Newton's Method, we have:

f(x) = 11eaa595_6603_dacd_a696_072577b78963_TB4584_11_TB4584_11

f'(x) = 5db5b424_8e37_8162_f2d9_9a53f8cf4a35_TB4584_11_TB4584_11

First iteration:

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

x1 = 11eaa595_6603_dace_a696_15cda0299fc9_TB4584_11_TB4584_11 - (f(11eaa595_6603_dace_a696_15cda0299fc9_TB4584_11_TB4584_11))/(f'(11eaa595_6603_dace_a696_15cda0299fc9_TB4584_11_TB4584_11))

x1 = 11eaa595_6604_0306_a694_f2d9_9a53f8cf4a35_TB4584_11_TB4584_11

Second iteration:

x2 = x1 - f(x1)/f'(x1)

x2 = 11eaa595_6604_0306_a694_f2d9_9a53f8cf4a35_TB4584_11_TB4584_11 - (f(11eaa595_6604_0306_a694_f2d9_9a53f8cf4a35_TB4584_11_TB4584_11))/(f'(11eaa595_6604_0306_a694_f2d9_9a53f8cf4a35_TB4584_11_TB4584_11))

x2 = 11eaa595_6604_7712_a696_172a925d890a_TB4584_11_TB4584_11

We can check if this value results in a close proximity to zero:

f(x2) = 1.0377 x 10^-9

Since f(x2) is very close to zero, this value is an acceptable approximation for the root of the function. Therefore, the best choice is D as it is the result of two iterations of Newton's Method with the given function and initial value.

Complete Two Iterations of Newton's Method for the Function