泰勒 sin(x), pi/6

解答

泰勒

sin (x), \frac{π}{6}

\frac{1}{2} + \frac{3}{2} (x - \frac{π}{6}) - \frac{1}{4} (x - \frac{π}{6})^{2} - \frac{1}{4 3} (x - \frac{π}{6})^{3} + \frac{1}{48} (x - \frac{π}{6})^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{2} + \frac{\frac{d}{d x} ( sin ( x ) ) ( \frac{π}{6} )}{1 !} (x - \frac{π}{6}) + \frac{\frac{d ^{2}}{d x ^{2}} ( sin ( x ) ) ( \frac{π}{6} )}{2 !} (x - \frac{π}{6})^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( sin ( x ) ) ( \frac{π}{6} )}{3 !} (x - \frac{π}{6})^{3} + \dots

求导数

= \frac{1}{2} + \frac{\frac{3}{2}}{1 !} (x - \frac{π}{6}) + \frac{- \frac{1}{2}}{2 !} (x - \frac{π}{6})^{2} + \frac{- \frac{3}{2}}{3 !} (x - \frac{π}{6})^{3} + \frac{\frac{1}{2}}{4 !} (x - \frac{π}{6})^{4} + \dots

整理后得

= \frac{1}{2} + \frac{3}{2} (x - \frac{π}{6}) - \frac{1}{4} (x - \frac{π}{6})^{2} - \frac{1}{4 3} (x - \frac{π}{6})^{3} + \frac{1}{48} (x - \frac{π}{6})^{4} + \dots

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