taylor x*cos(x),1

Lösung

taylor

x \cdot cos (x), 1

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

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

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

Fasse zusammen

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

x \to 3 lim (\frac{x ^{2} - 9}{3})

t an g e n t y = 11 - 6 e^{x}, (0, 5)

x \to 0 lim (\frac{ln ( x )}{x - 1})

\int 21 arctan (x) d x

x \to 0 lim (\frac{e ^{x} - 1}{4 x})