Newtons method question

Question

Use Newtons method to approximate √10 to 6 significant figures starting with x0=3.5.

Answer 1

Let f(x) = x² - 10 [ note that the roots of this function are ±√10.

f'(x) = 2x

N(x) = x - f(x)/f'(x)

N(7/2) = 7/2 - [(49/4 - 10)/7] = 7/2 - 9/28 = 89/28 ≈ 3.17857142857

N(89/28) or N(3.17857142857) = 89/28 - [(89/28)²-10)/(89/14)] and so on.

Probably want a calculator from here on out (until the 6th decimal digits start repeating).

desmos.com/calculator/bwnk2utfxt