泰勒 sin(x-pi/4)

解答

泰勒

sin (x - \frac{π}{4})

- \frac{2}{2} + \frac{1}{2} x + \frac{1}{2 2} x^{2} - \frac{1}{6 2} x^{3} - \frac{1}{24 2} x^{4} + \dots

求解步骤

应用泰勒公式

= - \frac{2}{2} + \frac{\frac{d}{d x} ( sin ( x - \frac{π}{4} ) ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( sin ( x - \frac{π}{4} ) ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( sin ( x - \frac{π}{4} ) ) ( 0 )}{3 !} x^{3} + \dots

求导数

= - \frac{2}{2} + \frac{\frac{2}{2}}{1 !} x + \frac{\frac{2}{2}}{2 !} x^{2} + \frac{- \frac{2}{2}}{3 !} x^{3} + \frac{- \frac{2}{2}}{4 !} x^{4} + \dots

整理后得

= - \frac{2}{2} + \frac{1}{2} x + \frac{1}{2 2} x^{2} - \frac{1}{6 2} x^{3} - \frac{1}{24 2} x^{4} + \dots