泰勒 (-3x+28)^{5/4}

解答

泰勒

(- 3 x + 28)^{\frac{5}{4}}

28 \cdot 2 8^{\frac{1}{4}} - \frac{15 \cdot 2 8 ^{\frac{1}{4}}}{4} x + \frac{45}{32 \cdot 2 8 ^{\frac{3}{4}}} x^{2} + \frac{135}{3584 \cdot 2 8 ^{\frac{3}{4}}} x^{3} + \frac{405}{458752 \cdot 7 ^{\frac{3}{4}} \cdot 2 ^{\frac{1}{2}}} x^{4} + \dots

求解步骤

应用泰勒公式

= 28 \cdot 2 8^{\frac{1}{4}} + \frac{\frac{d}{d x} ( ( - 3 x + 28 ) ^{\frac{5}{4}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ( - 3 x + 28 ) ^{\frac{5}{4}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ( - 3 x + 28 ) ^{\frac{5}{4}} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 28 \cdot 2 8^{\frac{1}{4}} + \frac{- \frac{15 \cdot 2 8 ^{\frac{1}{4}}}{4}}{1 !} x + \frac{\frac{45}{16 \cdot 2 8 ^{\frac{3}{4}}}}{2 !} x^{2} + \frac{\frac{405}{1792 \cdot 2 8 ^{\frac{3}{4}}}}{3 !} x^{3} + \frac{\frac{1215}{57344 \cdot 7 ^{\frac{3}{4}} \cdot 2 ^{\frac{1}{2}}}}{4 !} x^{4} + \dots

整理后得

= 28 \cdot 2 8^{\frac{1}{4}} - \frac{15 \cdot 2 8 ^{\frac{1}{4}}}{4} x + \frac{45}{32 \cdot 2 8 ^{\frac{3}{4}}} x^{2} + \frac{135}{3584 \cdot 2 8 ^{\frac{3}{4}}} x^{3} + \frac{405}{458752 \cdot 7 ^{\frac{3}{4}} \cdot 2 ^{\frac{1}{2}}} x^{4} + \dots

\int_{0}^{7} \frac{1}{49 + x ^{2}} d x

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x \to 3 - lim (\frac{7 - x - 4}{x - 3})

(1 + x^{14}) (\frac{d y}{d x}) + 14 x^{13} y = 11 x^{5}

t an g e n t 3^{x^{2} - 1}, at x = 2