taylor (sin(x)-x)/(x^2)

Lösung

taylor

\frac{sin ( x ) - x}{x ^{2}}

sin (1) - 1 + (1 + cos (1) - 2 sin (1)) (x - 1) + \frac{5 sin ( 1 ) - 4 cos ( 1 ) - 2}{2} (x - 1)^{2} + \frac{17 cos ( 1 ) - 18 sin ( 1 ) + 6}{6} (x - 1)^{3} + \frac{85 sin ( 1 ) - 88 cos ( 1 ) - 24}{24} (x - 1)^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= sin (1) - 1 + \frac{1 + cos ( 1 ) - 2 sin ( 1 )}{1 !} (x - 1) + \frac{5 sin ( 1 ) - 4 cos ( 1 ) - 2}{2 !} (x - 1)^{2} + \frac{17 cos ( 1 ) - 18 sin ( 1 ) + 6}{3 !} (x - 1)^{3} + \frac{85 sin ( 1 ) - 88 cos ( 1 ) - 24}{4 !} (x - 1)^{4} + \dots

Fasse zusammen

= sin (1) - 1 + (1 + cos (1) - 2 sin (1)) (x - 1) + \frac{5 sin ( 1 ) - 4 cos ( 1 ) - 2}{2} (x - 1)^{2} + \frac{17 cos ( 1 ) - 18 sin ( 1 ) + 6}{6} (x - 1)^{3} + \frac{85 sin ( 1 ) - 88 cos ( 1 ) - 24}{24} (x - 1)^{4} + \dots

x \to 0 lim (sin (\frac{1}{∣ x ∣}))

d er i v a t i v e \frac{3 x ^{2}}{4}

\frac{d}{d x} (2 + \frac{1}{x ^{2}})

\frac{\partial}{\partial x} (3 x^{2} + 2 y)

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