泰勒 e^{1/(x-9)},7

解答

泰勒

e^{\frac{1}{x - 9}}, 7

\frac{1}{e ^{\frac{1}{2}}} - \frac{1}{4 e ^{\frac{1}{2}}} (x - 7) - \frac{3}{32 e ^{\frac{1}{2}}} (x - 7)^{2} - \frac{13}{384 e ^{\frac{1}{2}}} (x - 7)^{3} - \frac{71}{6144 e ^{\frac{1}{2}}} (x - 7)^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{e ^{\frac{1}{2}}} + \frac{\frac{d}{d x} ( e ^{\frac{1}{x - 9}} ) ( 7 )}{1 !} (x - 7) + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{\frac{1}{x - 9}} ) ( 7 )}{2 !} (x - 7)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{\frac{1}{x - 9}} ) ( 7 )}{3 !} (x - 7)^{3} + \dots

求导数

= \frac{1}{e ^{\frac{1}{2}}} + \frac{- \frac{1}{4 e ^{\frac{1}{2}}}}{1 !} (x - 7) + \frac{- \frac{3}{16 e ^{\frac{1}{2}}}}{2 !} (x - 7)^{2} + \frac{- \frac{13}{64 e ^{\frac{1}{2}}}}{3 !} (x - 7)^{3} + \frac{- \frac{71}{256 e ^{\frac{1}{2}}}}{4 !} (x - 7)^{4} + \dots

整理后得

= \frac{1}{e ^{\frac{1}{2}}} - \frac{1}{4 e ^{\frac{1}{2}}} (x - 7) - \frac{3}{32 e ^{\frac{1}{2}}} (x - 7)^{2} - \frac{13}{384 e ^{\frac{1}{2}}} (x - 7)^{3} - \frac{71}{6144 e ^{\frac{1}{2}}} (x - 7)^{4} + \dots

\int \frac{x}{( 1 + 3 x )} d x

\int a^{x} d x

\frac{d}{d x} ((sin (3 x))^{4})

\int \frac{1}{2 - x} d x

x \to 0 lim (\frac{e ^{- 3 x} - 1}{x})