泰勒 cos(3x),2.5

解答

泰勒

cos (3 x), 2.5

0.34663 \dots - 2.81399 \dots (x - 2.5) - 1.55985 \dots (x - 2.5)^{2} + 4.22099 \dots (x - 2.5)^{3} + 1.16989 \dots (x - 2.5)^{4} + \dots

求解步骤

应用泰勒公式

= 0.34663 \dots + \frac{\frac{d}{d x} ( cos ( 3 x ) ) ( 2.5 )}{1 !} (x - 2.5) + \frac{\frac{d ^{2}}{d x ^{2}} ( cos ( 3 x ) ) ( 2.5 )}{2 !} (x - 2.5)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( cos ( 3 x ) ) ( 2.5 )}{3 !} (x - 2.5)^{3} + \dots

求导数

= 0.34663 \dots + \frac{- 2.81399 \dots}{1 !} (x - 2.5) + \frac{- 3.11971 \dots}{2 !} (x - 2.5)^{2} + \frac{25.32599 \dots}{3 !} (x - 2.5)^{3} + \frac{28.07746 \dots}{4 !} (x - 2.5)^{4} + \dots

整理后得

= 0.34663 \dots - 2.81399 \dots (x - 2.5) - 1.55985 \dots (x - 2.5)^{2} + 4.22099 \dots (x - 2.5)^{3} + 1.16989 \dots (x - 2.5)^{4} + \dots

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