テイラー sqrt(x),\at 4

解

テイラー

x, at 4

\frac{\frac{1}{2 at 4}}{1 !} (x - at 4) + \frac{- \frac{1}{4 ( at 4 ) ^{\frac{3}{2}}}}{2 !} (x - at 4)^{2} + \frac{\frac{3}{8 ( at 4 ) ^{\frac{5}{2}}}}{3 !} (x - at 4)^{3} + \frac{- \frac{15}{16 ( at 4 ) ^{\frac{7}{2}}}}{4 !} (x - at 4)^{4} + \dots

解答ステップ

テイラーの公式を適用する

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

導関数を評価する

= at 4 + \frac{\frac{1}{2 at 4}}{1 !} (x - at 4) + \frac{- \frac{1}{4 ( at 4 ) ^{\frac{3}{2}}}}{2 !} (x - at 4)^{2} + \frac{\frac{3}{8 ( at 4 ) ^{\frac{5}{2}}}}{3 !} (x - at 4)^{3} + \frac{- \frac{15}{16 ( at 4 ) ^{\frac{7}{2}}}}{4 !} (x - at 4)^{4} + \dots

改良

= \frac{\frac{1}{2 at 4}}{1 !} (x - at 4) + \frac{- \frac{1}{4 ( at 4 ) ^{\frac{3}{2}}}}{2 !} (x - at 4)^{2} + \frac{\frac{3}{8 ( at 4 ) ^{\frac{5}{2}}}}{3 !} (x - at 4)^{3} + \frac{- \frac{15}{16 ( at 4 ) ^{\frac{7}{2}}}}{4 !} (x - at 4)^{4} + \dots