泰勒 1/(2+3x)

解答

泰勒

\frac{1}{2 + 3 x}

\frac{1}{2} - \frac{3}{4} x + \frac{9}{8} x^{2} - \frac{27}{16} x^{3} + \frac{81}{32} x^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= \frac{1}{2} + \frac{- \frac{3}{4}}{1 !} x + \frac{\frac{9}{4}}{2 !} x^{2} + \frac{- \frac{81}{8}}{3 !} x^{3} + \frac{\frac{243}{4}}{4 !} x^{4} + \dots

整理后得

= \frac{1}{2} - \frac{3}{4} x + \frac{9}{8} x^{2} - \frac{27}{16} x^{3} + \frac{81}{32} x^{4} + \dots

y^{^{'}} = y + x, y (0) = - 1

\int (- 0.5 x + 450) d x

\frac{d y}{d x} = 4 x - 5

x \to 0 lim ((x)^{t a n (x)})

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