泰勒 1/(x^2),\at 4

解答

泰勒

\frac{1}{x ^{2}}, at 4

\frac{- \frac{2}{( at 4 ) ^{3}}}{1 !} (x - at 4) + \frac{\frac{6}{( at 4 ) ^{4}}}{2 !} (x - at 4)^{2} + \frac{- \frac{24}{( at 4 ) ^{5}}}{3 !} (x - at 4)^{3} + \frac{\frac{120}{( at 4 ) ^{6}}}{4 !} (x - at 4)^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{( at 4 ) ^{2}} + \frac{\frac{d}{d x} ( \frac{1}{x ^{2}} ) ( at 4 )}{1 !} (x - at 4) + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{x ^{2}} ) ( at 4 )}{2 !} (x - at 4)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{x ^{2}} ) ( at 4 )}{3 !} (x - at 4)^{3} + \dots

求导数

= \frac{1}{( at 4 ) ^{2}} + \frac{- \frac{2}{( at 4 ) ^{3}}}{1 !} (x - at 4) + \frac{\frac{6}{( at 4 ) ^{4}}}{2 !} (x - at 4)^{2} + \frac{- \frac{24}{( at 4 ) ^{5}}}{3 !} (x - at 4)^{3} + \frac{\frac{120}{( at 4 ) ^{6}}}{4 !} (x - at 4)^{4} + \dots

整理后得

= \frac{- \frac{2}{( at 4 ) ^{3}}}{1 !} (x - at 4) + \frac{\frac{6}{( at 4 ) ^{4}}}{2 !} (x - at 4)^{2} + \frac{- \frac{24}{( at 4 ) ^{5}}}{3 !} (x - at 4)^{3} + \frac{\frac{120}{( at 4 ) ^{6}}}{4 !} (x - at 4)^{4} + \dots

\frac{\partial}{\partial x} (e^{x} x + 2 e^{x} y)

\int_{2}^{3} 3 - x d x

y^{^{'}} = - 5 sin (3 t) + 2 cos (4 t)

x \to 0 lim (\frac{1}{sin ( 5 x )})

a re a y = - x^{2} + 16, x = - 2, y = 0, x = 3