taylor sin(1/x)

Lösung

taylor

sin (\frac{1}{x})

sin (1) - cos (1) (x - 1) + \frac{- sin ( 1 ) + 2 cos ( 1 )}{2} (x - 1)^{2} + \frac{- 5 cos ( 1 ) + 6 sin ( 1 )}{6} (x - 1)^{3} + \frac{12 cos ( 1 ) - 35 sin ( 1 )}{24} (x - 1)^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= sin (1) + \frac{- cos ( 1 )}{1 !} (x - 1) + \frac{- sin ( 1 ) + 2 cos ( 1 )}{2 !} (x - 1)^{2} + \frac{- 5 cos ( 1 ) + 6 sin ( 1 )}{3 !} (x - 1)^{3} + \frac{12 cos ( 1 ) - 35 sin ( 1 )}{4 !} (x - 1)^{4} + \dots

Fasse zusammen

= sin (1) - cos (1) (x - 1) + \frac{- sin ( 1 ) + 2 cos ( 1 )}{2} (x - 1)^{2} + \frac{- 5 cos ( 1 ) + 6 sin ( 1 )}{6} (x - 1)^{3} + \frac{12 cos ( 1 ) - 35 sin ( 1 )}{24} (x - 1)^{4} + \dots

x \to 0 + lim (\frac{ln ( x )}{x})

y^{^{''}} + 2 y^{^{'}} + 5 y = 4 cos (2 t) e^{- t}

\int \frac{1}{cos ^{5} ( x )} d x

d er i v a t i v e \frac{2}{x ^{\frac{1}{3}}} - 2

2 y^{^{'}} + t y = 7