泰勒 sqrt(x),100

解答

泰勒

x, 100

10 + \frac{1}{20} (x - 100) - \frac{1}{8000} (x - 100)^{2} + \frac{1}{1600000} (x - 100)^{3} - \frac{1}{256000000} (x - 100)^{4} + \dots

求解步骤

应用泰勒公式

= 10 + \frac{\frac{d}{d x} ( x ) ( 100 )}{1 !} (x - 100) + \frac{\frac{d ^{2}}{d x ^{2}} ( x ) ( 100 )}{2 !} (x - 100)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x ) ( 100 )}{3 !} (x - 100)^{3} + \dots

求导数

= 10 + \frac{\frac{1}{20}}{1 !} (x - 100) + \frac{- \frac{1}{4000}}{2 !} (x - 100)^{2} + \frac{\frac{3}{800000}}{3 !} (x - 100)^{3} + \frac{- \frac{3}{32000000}}{4 !} (x - 100)^{4} + \dots

整理后得

= 10 + \frac{1}{20} (x - 100) - \frac{1}{8000} (x - 100)^{2} + \frac{1}{1600000} (x - 100)^{3} - \frac{1}{256000000} (x - 100)^{4} + \dots

\frac{d y}{d x} = x + 2 y, y (0) = 7

\frac{d}{d x} (e^{x} \cdot cos (y))

\int y e^{y} d y

x \to 0 lim (\frac{x ^{2} + 1}{x})

\frac{d}{d x} (- x^{3} + 5 x^{2} - 8 x)