泰勒 e^{-7x}

解答

泰勒

e^{- 7 x}

1 - 7 x + \frac{49}{2} x^{2} - \frac{343}{6} x^{3} + \frac{2401}{24} x^{4} + \dots

求解步骤

应用泰勒公式

= 1 + \frac{\frac{d}{d x} ( e ^{- 7 x} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{- 7 x} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{- 7 x} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 1 + \frac{- 7}{1 !} x + \frac{49}{2 !} x^{2} + \frac{- 343}{3 !} x^{3} + \frac{2401}{4 !} x^{4} + \dots

整理后得

= 1 - 7 x + \frac{49}{2} x^{2} - \frac{343}{6} x^{3} + \frac{2401}{24} x^{4} + \dots

d er i v a t i v e y = x^{3} (8 x + 12)

f^{^{'}} (x) = 1 - 3 x

\frac{\partial}{\partial x} (\frac{1}{y} + x ln (y))

s l o p e (- 1, - 1), (4, - 4)

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