泰勒 f(x)=sin(x),0

解答

泰勒

f (x) = sin (x), 0

求解步骤

应用泰勒公式

= 0 + \frac{\frac{d}{d x} ( sin ( x ) ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( sin ( x ) ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( sin ( x ) ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 1}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{1}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- 1}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{1}{9 !} x^{9} + \dots

整理后得

= \frac{1}{1 !} x - \frac{1}{3 !} x^{3} + \frac{1}{5 !} x^{5} - \frac{1}{7 !} x^{7} + \frac{1}{9 !} x^{9} + \dots

级数求和的表示形式

= n = 0 \sum \infty (- 1)^{n} \frac{x ^{2 n + 1}}{( 2 n + 1 ) !}

a re a x^{2} + x + 3, [1, 3]

\int \frac{2}{3} cos^{2} (3 x) d x

ma c l a u r in \frac{1}{1 + 4 x ^{2}}

d er i v a t i v e f (x) = e^{x} - cot (x)

\frac{d}{d x} (x sin (x^{3}))