泰勒 8x^4-1x^2+9x+2,x=0.05

解答

泰勒

8 x^{4} - 1 x^{2} + 9 x + 2, x = 0.05

2.44755 + 8.904 (x - 0.05) - 0.88 (x - 0.05)^{2} + 1.6 (x - 0.05)^{3} + 8 (x - 0.05)^{4}

求解步骤

应用泰勒公式

= 2.44755 + \frac{\frac{d}{d x} ( 8 x ^{4} - 1 \cdot x ^{2} + 9 x + 2 ) ( 0.05 )}{1 !} (x - 0.05) + \frac{\frac{d ^{2}}{d x ^{2}} ( 8 x ^{4} - 1 \cdot x ^{2} + 9 x + 2 ) ( 0.05 )}{2 !} (x - 0.05)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( 8 x ^{4} - 1 \cdot x ^{2} + 9 x + 2 ) ( 0.05 )}{3 !} (x - 0.05)^{3} + \dots

求导数

= 2.44755 + \frac{8.904}{1 !} (x - 0.05) + \frac{- 1.76}{2 !} (x - 0.05)^{2} + \frac{9.6}{3 !} (x - 0.05)^{3} + \frac{192}{4 !} (x - 0.05)^{4}

整理后得

= 2.44755 + 8.904 (x - 0.05) - 0.88 (x - 0.05)^{2} + 1.6 (x - 0.05)^{3} + 8 (x - 0.05)^{4}

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