泰勒 tan(x),0,1,2,3, pi/4

解答

泰勒

tan (x), 0, 1, 2, 3, \frac{π}{4}

x + \frac{1}{3} x^{3} + \frac{2}{15} x^{5} + \frac{17}{315} x^{7} + \frac{62}{2835} x^{9} + \dots

求解步骤

应用泰勒公式

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

求导数

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{2}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{16}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{272}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{7936}{9 !} x^{9} + \dots

整理后得

= x + \frac{1}{3} x^{3} + \frac{2}{15} x^{5} + \frac{17}{315} x^{7} + \frac{62}{2835} x^{9} + \dots

a re a y = 7 x, y = 6 x^{2}

a re a x = 0, y = 2, y = x

(sin (t) - t cos (t))^{^{'}}

\frac{d}{d x} (\frac{4 x}{x + 5})

\frac{d}{d x} ((1 - 3 x)^{2} (x^{2} - 2)^{3})