泰勒 sin(3x), pi/3

解答

泰勒

sin (3 x), \frac{π}{3}

- 3 (x - \frac{π}{3}) + \frac{9}{2} (x - \frac{π}{3})^{3} - \frac{81}{40} (x - \frac{π}{3})^{5} + \frac{243}{560} (x - \frac{π}{3})^{7} - \frac{243}{4480} (x - \frac{π}{3})^{9} + \dots

求解步骤

应用泰勒公式

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

求导数

= 0 + \frac{- 3}{1 !} (x - \frac{π}{3}) + \frac{0}{2 !} (x - \frac{π}{3})^{2} + \frac{27}{3 !} (x - \frac{π}{3})^{3} + \frac{0}{4 !} (x - \frac{π}{3})^{4} + \frac{- 243}{5 !} (x - \frac{π}{3})^{5} + \frac{0}{6 !} (x - \frac{π}{3})^{6} + \frac{2187}{7 !} (x - \frac{π}{3})^{7} + \frac{0}{8 !} (x - \frac{π}{3})^{8} + \frac{- 19683}{9 !} (x - \frac{π}{3})^{9} + \dots

整理后得

= - 3 (x - \frac{π}{3}) + \frac{9}{2} (x - \frac{π}{3})^{3} - \frac{81}{40} (x - \frac{π}{3})^{5} + \frac{243}{560} (x - \frac{π}{3})^{7} - \frac{243}{4480} (x - \frac{π}{3})^{9} + \dots

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