PART 11：例題-求近似值

利用牛頓求根法求 \(\sqrt 3 \) 至小數第3位

SOL:  \(x = \sqrt 3   \Rightarrow {x^2} = 3\) \( \Rightarrow f(x) = {x^2} - 3\) ， \( \Rightarrow f'(x) = 2x\)

(1)令 \({x_0} = 1.7\) \( \Rightarrow {x_1} = 1.7 - \frac{{2.25 - 3}}{3} = 1.7 - \frac{{ - 0.11}}{{3.4}} = 1.73235\)

(2)令 \({x_1} = 1.73235\) \(  \Rightarrow {x_2} = 1.73235 - \frac{{2.25 - 3}}{3} = 1.73235 - \frac{{0.001036}}{{3.4647}} = 1.732050\)

\( \sqrt 3  \buildrel\textstyle.\over= 1.732\)