taylor sin(2x), pi/4

Lösung

taylor

sin (2 x), \frac{π}{4}

1 - 2 (x - \frac{π}{4})^{2} + \frac{2}{3} (x - \frac{π}{4})^{4} - \frac{4}{45} (x - \frac{π}{4})^{6} + \frac{2}{315} (x - \frac{π}{4})^{8} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} (x - \frac{π}{4}) + \frac{- 4}{2 !} (x - \frac{π}{4})^{2} + \frac{0}{3 !} (x - \frac{π}{4})^{3} + \frac{16}{4 !} (x - \frac{π}{4})^{4} + \frac{0}{5 !} (x - \frac{π}{4})^{5} + \frac{- 64}{6 !} (x - \frac{π}{4})^{6} + \frac{0}{7 !} (x - \frac{π}{4})^{7} + \frac{256}{8 !} (x - \frac{π}{4})^{8} + \dots

Fasse zusammen

= 1 - 2 (x - \frac{π}{4})^{2} + \frac{2}{3} (x - \frac{π}{4})^{4} - \frac{4}{45} (x - \frac{π}{4})^{6} + \frac{2}{315} (x - \frac{π}{4})^{8} + \dots