泰勒 sin(x)(cos(x)-1)pi

解答

泰勒

sin (x) (cos (x) - 1) π

- \frac{π}{2} x^{3} + \frac{π}{8} x^{5} - \frac{π}{80} x^{7} + \frac{17 π}{24192} x^{9} + \dots

求解步骤

应用泰勒公式

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

求导数

= 0 + \frac{0}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 3 π}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{15 π}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- 63 π}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{255 π}{9 !} x^{9} + \frac{0}{10 !} x^{10} + \dots

整理后得

= - \frac{π}{2} x^{3} + \frac{π}{8} x^{5} - \frac{π}{80} x^{7} + \frac{17 π}{24192} x^{9} + \dots

\frac{\partial}{\partial x} (x^{2} + 4 y^{2})

x^{2} y^{^{''}} - x y^{^{'}} + y = \frac{1}{x}

\int_{0}^{0.5} 500 x d x

\int_{0}^{16} x e^{- x} d x

x \to 0 - lim (\frac{11}{tan ( x )})