泰勒 (-5x+24)^{3/2}

解答

泰勒

(- 5 x + 24)^{\frac{3}{2}}

48 \cdot 6^{\frac{1}{2}} - 15 \cdot 6^{\frac{1}{2}} x + \frac{25 \cdot 3 ^{\frac{1}{2}}}{16 \cdot 2 ^{\frac{1}{2}}} x^{2} + \frac{125}{768 \cdot 6 ^{\frac{1}{2}}} x^{3} + \frac{625}{49152 \cdot 6 ^{\frac{1}{2}}} x^{4} + \dots

求解步骤

应用泰勒公式

= 48 \cdot 6^{\frac{1}{2}} + \frac{\frac{d}{d x} ( ( - 5 x + 24 ) ^{\frac{3}{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ( - 5 x + 24 ) ^{\frac{3}{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ( - 5 x + 24 ) ^{\frac{3}{2}} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 48 \cdot 6^{\frac{1}{2}} + \frac{- 15 \cdot 6 ^{\frac{1}{2}}}{1 !} x + \frac{\frac{25 \cdot 3 ^{\frac{1}{2}}}{8 \cdot 2 ^{\frac{1}{2}}}}{2 !} x^{2} + \frac{\frac{125}{128 \cdot 6 ^{\frac{1}{2}}}}{3 !} x^{3} + \frac{\frac{625}{2048 \cdot 6 ^{\frac{1}{2}}}}{4 !} x^{4} + \dots

整理后得

= 48 \cdot 6^{\frac{1}{2}} - 15 \cdot 6^{\frac{1}{2}} x + \frac{25 \cdot 3 ^{\frac{1}{2}}}{16 \cdot 2 ^{\frac{1}{2}}} x^{2} + \frac{125}{768 \cdot 6 ^{\frac{1}{2}}} x^{3} + \frac{625}{49152 \cdot 6 ^{\frac{1}{2}}} x^{4} + \dots

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