泰勒 1/(x^2),7

解答

泰勒

\frac{1}{x ^{2}}, 7

\frac{1}{49} - \frac{2}{343} (x - 7) + \frac{3}{2401} (x - 7)^{2} - \frac{4}{16807} (x - 7)^{3} + \frac{5}{117649} (x - 7)^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{49} + \frac{\frac{d}{d x} ( \frac{1}{x ^{2}} ) ( 7 )}{1 !} (x - 7) + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{x ^{2}} ) ( 7 )}{2 !} (x - 7)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{x ^{2}} ) ( 7 )}{3 !} (x - 7)^{3} + \dots

求导数

= \frac{1}{49} + \frac{- \frac{2}{343}}{1 !} (x - 7) + \frac{\frac{6}{2401}}{2 !} (x - 7)^{2} + \frac{- \frac{24}{16807}}{3 !} (x - 7)^{3} + \frac{\frac{120}{117649}}{4 !} (x - 7)^{4} + \dots

整理后得

= \frac{1}{49} - \frac{2}{343} (x - 7) + \frac{3}{2401} (x - 7)^{2} - \frac{4}{16807} (x - 7)^{3} + \frac{5}{117649} (x - 7)^{4} + \dots

d er i v a t i v e x^{\frac{5}{2}} e^{x}

x \to 9 + lim (\frac{2}{x + 9})

d er i v a t i v e e^{x + 2} + 2

\int - \frac{7}{5} x^{- 12} d x

\frac{d}{d x} (e^{c o s (x)} sin (x^{2} - x))