泰勒 x^{-2},6

解答

泰勒

x^{- 2}, 6

\frac{1}{36} - \frac{1}{108} (x - 6) + \frac{1}{432} (x - 6)^{2} - \frac{1}{1944} (x - 6)^{3} + \frac{5}{46656} (x - 6)^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{36} + \frac{\frac{d}{d x} ( x ^{- 2} ) ( 6 )}{1 !} (x - 6) + \frac{\frac{d ^{2}}{d x ^{2}} ( x ^{- 2} ) ( 6 )}{2 !} (x - 6)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x ^{- 2} ) ( 6 )}{3 !} (x - 6)^{3} + \dots

求导数

= \frac{1}{36} + \frac{- \frac{1}{108}}{1 !} (x - 6) + \frac{\frac{1}{216}}{2 !} (x - 6)^{2} + \frac{- \frac{1}{324}}{3 !} (x - 6)^{3} + \frac{\frac{5}{1944}}{4 !} (x - 6)^{4} + \dots

整理后得

= \frac{1}{36} - \frac{1}{108} (x - 6) + \frac{1}{432} (x - 6)^{2} - \frac{1}{1944} (x - 6)^{3} + \frac{5}{46656} (x - 6)^{4} + \dots

a re a y^{2} = x^{3}, y^{2} = x

\int \frac{- 9 x ^{2} + 2}{x ^{3} + 9 x} d x

\int \frac{1}{x ^{2} x ^{2} + 49} d x

\frac{d}{d x} (\frac{2}{( 1 - \frac{1}{3} x ^{3} + 1 ) ^{2}})

d er i v a t i v e 5 x^{2} - 2 x - 1