泰勒 e^{-5x-3},5

解答

泰勒

e^{- 5 x - 3}, 5

\frac{1}{e ^{28}} - \frac{5}{e ^{28}} (x - 5) + \frac{25}{2 e ^{28}} (x - 5)^{2} - \frac{125}{6 e ^{28}} (x - 5)^{3} + \frac{625}{24 e ^{28}} (x - 5)^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{e ^{28}} + \frac{\frac{d}{d x} ( e ^{- 5 x - 3} ) ( 5 )}{1 !} (x - 5) + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{- 5 x - 3} ) ( 5 )}{2 !} (x - 5)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{- 5 x - 3} ) ( 5 )}{3 !} (x - 5)^{3} + \dots

求导数

= \frac{1}{e ^{28}} + \frac{- \frac{5}{e ^{28}}}{1 !} (x - 5) + \frac{\frac{25}{e ^{28}}}{2 !} (x - 5)^{2} + \frac{- \frac{125}{e ^{28}}}{3 !} (x - 5)^{3} + \frac{\frac{625}{e ^{28}}}{4 !} (x - 5)^{4} + \dots

整理后得

= \frac{1}{e ^{28}} - \frac{5}{e ^{28}} (x - 5) + \frac{25}{2 e ^{28}} (x - 5)^{2} - \frac{125}{6 e ^{28}} (x - 5)^{3} + \frac{625}{24 e ^{28}} (x - 5)^{4} + \dots

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