泰勒 (x^2+1/4)^{3/2}

解答

泰勒

(x^{2} + \frac{1}{4})^{\frac{3}{2}}

\frac{1}{8} + \frac{3}{4} x^{2} + \frac{3}{4} x^{4} - \frac{1}{2} x^{6} + \frac{3}{4} x^{8} + \dots

求解步骤

应用泰勒公式

= \frac{1}{8} + \frac{\frac{d}{d x} ( ( x ^{2} + \frac{1}{4} ) ^{\frac{3}{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ( x ^{2} + \frac{1}{4} ) ^{\frac{3}{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ( x ^{2} + \frac{1}{4} ) ^{\frac{3}{2}} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= \frac{1}{8} + \frac{0}{1 !} x + \frac{\frac{3}{2}}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{18}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 360}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{30240}{8 !} x^{8} + \dots

整理后得

= \frac{1}{8} + \frac{3}{4} x^{2} + \frac{3}{4} x^{4} - \frac{1}{2} x^{6} + \frac{3}{4} x^{8} + \dots

\int 8 x^{\frac{3}{5}} + 7 x^{- \frac{4}{5}} d x

l a pl a ce t r an s f or m 1 + e^{- t}

d er i v a t i v e \frac{4 x - 3}{x}

y^{^{''}} - \frac{3}{x} y^{^{'}} - \frac{4}{x ^{2}} y = 0

\frac{\partial}{\partial x} (17 x^{3} y^{2})