泰勒 (e^{x^2}-1)/(x^2)

解答

泰勒

\frac{e ^{x^{2}} - 1}{x ^{2}}

e - 1 + 2 (x - 1) + (2 e - 3) (x - 1)^{2} + \frac{2 ( - e + 6 )}{3} (x - 1)^{3} + \frac{15 e - 30}{6} (x - 1)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= e - 1 + \frac{2}{1 !} (x - 1) + \frac{2 ( 2 e - 3 )}{2 !} (x - 1)^{2} + \frac{4 ( - e + 6 )}{3 !} (x - 1)^{3} + \frac{4 ( 15 e - 30 )}{4 !} (x - 1)^{4} + \dots

整理后得

= e - 1 + 2 (x - 1) + (2 e - 3) (x - 1)^{2} + \frac{2 ( - e + 6 )}{3} (x - 1)^{3} + \frac{15 e - 30}{6} (x - 1)^{4} + \dots

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

t an g e n t y = \frac{2}{x}, (\frac{1}{16}, 8)

d er i v a t i v e \frac{3 x ^{6}}{cos ( x ) e ^{x}}

a re a x^{2}, \frac{1}{2} x^{2}, 3 x

t an g e n t f (x) = x^{2} - 3, (- 2, 1)