泰勒 e^{x-1},0

解答

泰勒

e^{x - 1}, 0

\frac{1}{e} + \frac{1}{e} x + \frac{1}{2 e} x^{2} + \frac{1}{6 e} x^{3} + \frac{1}{24 e} x^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{e} + \frac{\frac{d}{d x} ( e ^{x - 1} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{x - 1} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{x - 1} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= \frac{1}{e} + \frac{\frac{1}{e}}{1 !} x + \frac{\frac{1}{e}}{2 !} x^{2} + \frac{\frac{1}{e}}{3 !} x^{3} + \frac{\frac{1}{e}}{4 !} x^{4} + \dots

整理后得

= \frac{1}{e} + \frac{1}{e} x + \frac{1}{2 e} x^{2} + \frac{1}{6 e} x^{3} + \frac{1}{24 e} x^{4} + \dots

y + sin (x) = x^{3} y^{^{'}}

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