如何用微分求0.956^1/(5)的近似值

题目

如何用微分求0.956^1/(5)的近似值

答案

解答:
用二项式展开估计:
(0.956)^(1/5)
= (1 - 0.044)^(1/5)
= 1 - C(1/5,1)×0.044 + C(1/5,2)×0.044² - C(1/5,3)×0.044³ + C(1/5,4)×0.044⁴ - .
= 1 - 0.0088 - 0.000154882 + 0.000004088832.. - 0.000000125936 .
= 0.991049 (6 s.f. 六位有效数)