泰勒 x^2cos(x),pi

解答

泰勒

x^{2} cos (x), π

- π^{2} - 2 π (x - π) + \frac{π ^{2} - 2}{2} (x - π)^{2} + π (x - π)^{3} + \frac{- π ^{2} + 12}{24} (x - π)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= - π^{2} + \frac{- 2 π}{1 !} (x - π) + \frac{π ^{2} - 2}{2 !} (x - π)^{2} + \frac{6 π}{3 !} (x - π)^{3} + \frac{- π ^{2} + 12}{4 !} (x - π)^{4} + \dots

整理后得

= - π^{2} - 2 π (x - π) + \frac{π ^{2} - 2}{2} (x - π)^{2} + π (x - π)^{3} + \frac{- π ^{2} + 12}{24} (x - π)^{4} + \dots

\int \frac{1}{2 ^{x}} d x

d er i v a t i v e y = x^{2} sin^{9} (x) + x cos^{- 4} (x)

\int \frac{1}{x ^{2} 4 x ^{2} + 9} d x

d er i v a t i v e f (x) = 6 x^{3} - \frac{27}{2} x^{2} + 3

d er i v a t i v e \frac{8}{7 x ^{2}}