テイラー sin(x), pi/2

解

テイラー

sin (x), \frac{π}{2}

1 - \frac{1}{2} (x - \frac{π}{2})^{2} + \frac{1}{24} (x - \frac{π}{2})^{4} - \frac{1}{720} (x - \frac{π}{2})^{6} + \frac{1}{40320} (x - \frac{π}{2})^{8} + \dots

解答ステップ

テイラーの公式を適用する

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

導関数を評価する

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

改良

= 1 - \frac{1}{2} (x - \frac{π}{2})^{2} + \frac{1}{24} (x - \frac{π}{2})^{4} - \frac{1}{720} (x - \frac{π}{2})^{6} + \frac{1}{40320} (x - \frac{π}{2})^{8} + \dots