taylor 7sin(x), pi/4

Lösung

taylor

7 sin (x), \frac{π}{4}

\frac{7}{2} + \frac{7}{2} (x - \frac{π}{4}) - \frac{7}{2 2} (x - \frac{π}{4})^{2} - \frac{7}{6 2} (x - \frac{π}{4})^{3} + \frac{7}{24 2} (x - \frac{π}{4})^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

= \frac{7}{2} + \frac{\frac{d}{d x} ( 7 sin ( x ) ) ( \frac{π}{4} )}{1 !} (x - \frac{π}{4}) + \frac{\frac{d ^{2}}{d x ^{2}} ( 7 sin ( x ) ) ( \frac{π}{4} )}{2 !} (x - \frac{π}{4})^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( 7 sin ( x ) ) ( \frac{π}{4} )}{3 !} (x - \frac{π}{4})^{3} + \dots

Evaluate Derivatives

= \frac{7}{2} + \frac{\frac{7}{2}}{1 !} (x - \frac{π}{4}) + \frac{- \frac{7}{2}}{2 !} (x - \frac{π}{4})^{2} + \frac{- \frac{7}{2}}{3 !} (x - \frac{π}{4})^{3} + \frac{\frac{7}{2}}{4 !} (x - \frac{π}{4})^{4} + \dots

Fasse zusammen

= \frac{7}{2} + \frac{7}{2} (x - \frac{π}{4}) - \frac{7}{2 2} (x - \frac{π}{4})^{2} - \frac{7}{6 2} (x - \frac{π}{4})^{3} + \frac{7}{24 2} (x - \frac{π}{4})^{4} + \dots

x d y - y d x + (y^{2} - 1) d y = 0

\frac{7 y ^{2} - t ^{2}}{y ^{5}} \frac{d y}{d t} + \frac{t}{2 y ^{4}} = 0

x \to 1 lim (\frac{4}{x + 3})

d er i v a t i v e t^{2} (t + 1)^{- 1}

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