泰勒 cos(x), pi/2

解答

泰勒

cos (x), \frac{π}{2}

- (x - \frac{π}{2}) + \frac{1}{6} (x - \frac{π}{2})^{3} - \frac{1}{120} (x - \frac{π}{2})^{5} + \frac{1}{5040} (x - \frac{π}{2})^{7} - \frac{1}{362880} (x - \frac{π}{2})^{9} + \dots

求解步骤

应用泰勒公式

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

求导数

= 0 + \frac{- 1}{1 !} (x - \frac{π}{2}) + \frac{0}{2 !} (x - \frac{π}{2})^{2} + \frac{1}{3 !} (x - \frac{π}{2})^{3} + \frac{0}{4 !} (x - \frac{π}{2})^{4} + \frac{- 1}{5 !} (x - \frac{π}{2})^{5} + \frac{0}{6 !} (x - \frac{π}{2})^{6} + \frac{1}{7 !} (x - \frac{π}{2})^{7} + \frac{0}{8 !} (x - \frac{π}{2})^{8} + \frac{- 1}{9 !} (x - \frac{π}{2})^{9} + \dots

整理后得

= - (x - \frac{π}{2}) + \frac{1}{6} (x - \frac{π}{2})^{3} - \frac{1}{120} (x - \frac{π}{2})^{5} + \frac{1}{5040} (x - \frac{π}{2})^{7} - \frac{1}{362880} (x - \frac{π}{2})^{9} + \dots

\frac{\partial}{\partial y} (x^{4} y - 3 x^{5} y^{2})

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\int x ln (3 x) d x

\frac{d}{d x} (x (x - 15)^{2})

\int e^{s i n (x)} cos (x) d x