泰勒 \sqrt[4]{x},11

解答

泰勒

4 x, 11

411 + \frac{1}{4 \cdot 1 1 ^{\frac{3}{4}}} (x - 11) - \frac{3}{352 \cdot 1 1 ^{\frac{3}{4}}} (x - 11)^{2} + \frac{7}{15488 \cdot 1 1 ^{\frac{3}{4}}} (x - 11)^{3} - \frac{7}{247808 \cdot 1 1 ^{\frac{3}{4}}} (x - 11)^{4} + \dots

求解步骤

应用泰勒公式

= 411 + \frac{\frac{d}{d x} ( 4 x ) ( 11 )}{1 !} (x - 11) + \frac{\frac{d ^{2}}{d x ^{2}} ( 4 x ) ( 11 )}{2 !} (x - 11)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( 4 x ) ( 11 )}{3 !} (x - 11)^{3} + \dots

求导数

= 411 + \frac{\frac{1}{4 \cdot 1 1 ^{\frac{3}{4}}}}{1 !} (x - 11) + \frac{- \frac{3}{176 \cdot 1 1 ^{\frac{3}{4}}}}{2 !} (x - 11)^{2} + \frac{\frac{21}{7744 \cdot 1 1 ^{\frac{3}{4}}}}{3 !} (x - 11)^{3} + \frac{- \frac{21}{30976 \cdot 1 1 ^{\frac{3}{4}}}}{4 !} (x - 11)^{4} + \dots

整理后得

= 411 + \frac{1}{4 \cdot 1 1 ^{\frac{3}{4}}} (x - 11) - \frac{3}{352 \cdot 1 1 ^{\frac{3}{4}}} (x - 11)^{2} + \frac{7}{15488 \cdot 1 1 ^{\frac{3}{4}}} (x - 11)^{3} - \frac{7}{247808 \cdot 1 1 ^{\frac{3}{4}}} (x - 11)^{4} + \dots

x \to 4 lim (\frac{6}{( x - 4 ) ^{2}})

\frac{d y}{d x} = x y + 3 x + 2 y + 6

(y^{2} + 3 x y) d x - 3 x^{2} d y = 0

in v erse l a pl a ce 1 + s

t an g e n t f (x) = 23 sin (x) + 2 cos (x)