泰勒 sin(x), pi/4

解答

泰勒

sin (x), \frac{π}{4}

\frac{2}{2} + \frac{1}{2} (x - \frac{π}{4}) - \frac{1}{2 2} (x - \frac{π}{4})^{2} - \frac{1}{6 2} (x - \frac{π}{4})^{3} + \frac{1}{24 2} (x - \frac{π}{4})^{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} )}{1 !} (x - \frac{π}{4}) + \frac{\frac{d ^{2}}{d x ^{2}} ( sin ( x ) ) ( \frac{π}{4} )}{2 !} (x - \frac{π}{4})^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( sin ( x ) ) ( \frac{π}{4} )}{3 !} (x - \frac{π}{4})^{3} + \dots

求导数

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

整理后得

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

y^{^{''}} + 6 y^{^{'}} + 8 y = 4 sin (2 t)

\int \frac{x ^{2} + 1}{( x - 7 ) ( x - 6 ) ^{2}} d x

y^{^{'}} = - 0.5 y + x

\int \frac{y}{2} d y

d er i v a t i v e f (x) = 5 x 2