テイラー cos(x),\at pi/2

解

テイラー

cos (x), at \frac{π}{2}

\frac{- sin ( at \frac{π}{2} )}{1 !} (x - at \frac{π}{2}) + \frac{- cos ( at \frac{π}{2} )}{2 !} (x - at \frac{π}{2})^{2} + \frac{sin ( at \frac{π}{2} )}{3 !} (x - at \frac{π}{2})^{3} + \frac{cos ( at \frac{π}{2} )}{4 !} (x - at \frac{π}{2})^{4} + \dots

解答ステップ

テイラーの公式を適用する

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

導関数を評価する

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

改良

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