泰勒 2+t+1/t

解答

泰勒

2 + t + \frac{1}{t}

4 + (t - 1)^{2} - (t - 1)^{3} + (t - 1)^{4} - (t - 1)^{5} + \dots

求解步骤

应用泰勒公式

= 4 + \frac{\frac{d}{d t} ( 2 + t + \frac{1}{t} ) ( 1 )}{1 !} (t - 1) + \frac{\frac{d ^{2}}{d t ^{2}} ( 2 + t + \frac{1}{t} ) ( 1 )}{2 !} (t - 1)^{2} + \frac{\frac{d ^{3}}{d t ^{3}} ( 2 + t + \frac{1}{t} ) ( 1 )}{3 !} (t - 1)^{3} + \dots

求导数

= 4 + \frac{0}{1 !} (t - 1) + \frac{2}{2 !} (t - 1)^{2} + \frac{- 6}{3 !} (t - 1)^{3} + \frac{24}{4 !} (t - 1)^{4} + \frac{- 120}{5 !} (t - 1)^{5} + \dots

整理后得

= 4 + (t - 1)^{2} - (t - 1)^{3} + (t - 1)^{4} - (t - 1)^{5} + \dots

\int \frac{x + 1}{x ( x - 1 ) ^{2}} d x

\frac{d}{d x} (5 cos (5 x))

\frac{d}{d x} (\frac{x ( x + 1 )}{2 x - 1})

\int \frac{1}{x ( x + 1 )} d x

d er i v a t i v e y = 2 - x^{2}