PART 4：例題-開立方根

試以微分公式估計 \(\sqrt[3]{{26.98}}\) 之近似值

SOL:

(1)設 \(f(x) = \sqrt[3]{x}\) ， \(f'(x) = \frac{1}{{3\sqrt[3]{{{x^2}}}}}\)

(2)微分公式 \(f(x + \Delta x) \buildrel\textstyle.\over= f(x) + f'(x)\Delta x\)

(3)令 \(x = 27\) ， \(\Delta x =  - 0.02\)

\(f(27 - 0.02) \buildrel\textstyle.\over= f(27) + f'(27)\left( { - 0.02} \right)\)

\(f(26.98) \buildrel\textstyle.\over= 3 - \frac{1}{3} \cdot \frac{1}{9}\left( {0.02} \right)\) \( = 3 - 0.00074\)  \(= 2.99926\)