泰勒 xcos(2x),1

解答

泰勒

x cos (2 x), 1

cos (2) + (cos (2) - 2 sin (2)) (x - 1) + \frac{- 4 sin ( 2 ) - 4 cos ( 2 )}{2} (x - 1)^{2} + \frac{- 12 cos ( 2 ) + 8 sin ( 2 )}{6} (x - 1)^{3} + \frac{32 sin ( 2 ) + 16 cos ( 2 )}{24} (x - 1)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= cos (2) + \frac{cos ( 2 ) - 2 sin ( 2 )}{1 !} (x - 1) + \frac{- 4 sin ( 2 ) - 4 cos ( 2 )}{2 !} (x - 1)^{2} + \frac{- 12 cos ( 2 ) + 8 sin ( 2 )}{3 !} (x - 1)^{3} + \frac{32 sin ( 2 ) + 16 cos ( 2 )}{4 !} (x - 1)^{4} + \dots

整理后得

= cos (2) + (cos (2) - 2 sin (2)) (x - 1) + \frac{- 4 sin ( 2 ) - 4 cos ( 2 )}{2} (x - 1)^{2} + \frac{- 12 cos ( 2 ) + 8 sin ( 2 )}{6} (x - 1)^{3} + \frac{32 sin ( 2 ) + 16 cos ( 2 )}{24} (x - 1)^{4} + \dots

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