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泰勒 x^3,3

解答

泰勒

x^{3}, 3

解答

27 + 27 (x - 3) + 9 (x - 3)^{2} + (x - 3)^{3}

求解步骤

应用泰勒公式

= 27 + \frac{\frac{d}{d x} ( x ^{3} ) ( 3 )}{1 !} (x - 3) + \frac{\frac{d ^{2}}{d x ^{2}} ( x ^{3} ) ( 3 )}{2 !} (x - 3)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x ^{3} ) ( 3 )}{3 !} (x - 3)^{3} + \dots

求导数

= 27 + \frac{27}{1 !} (x - 3) + \frac{18}{2 !} (x - 3)^{2} + \frac{6}{3 !} (x - 3)^{3} + \frac{0}{4 !} (x - 3)^{4} + \frac{0}{5 !} (x - 3)^{5} + \frac{0}{6 !} (x - 3)^{6} + \frac{0}{7 !} (x - 3)^{7} + \frac{0}{8 !} (x - 3)^{8} + \frac{0}{9 !} (x - 3)^{9} + \frac{0}{10 !} (x - 3)^{10}

整理后得

= 27 + 27 (x - 3) + 9 (x - 3)^{2} + (x - 3)^{3}

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