taylor cos(pix) 1/2

Lösung

taylor

cos (π x) \frac{1}{2}

\frac{1}{2} - \frac{π ^{2}}{4} x^{2} + \frac{π ^{4}}{48} x^{4} - \frac{π ^{6}}{1440} x^{6} + \frac{π ^{8}}{80640} x^{8} + \dots

Schritte zur Lösung

Apply the Taylor Formula

= \frac{1}{2} + \frac{\frac{d}{d x} ( cos ( π x ) \frac{1}{2} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( cos ( π x ) \frac{1}{2} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( cos ( π x ) \frac{1}{2} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

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

Fasse zusammen

= \frac{1}{2} - \frac{π ^{2}}{4} x^{2} + \frac{π ^{4}}{48} x^{4} - \frac{π ^{6}}{1440} x^{6} + \frac{π ^{8}}{80640} x^{8} + \dots

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

(x + 2 y) d x + (2 x - 3 y) d y = 0

\frac{d ^{4} y}{d x ^{4}} - 3 y = 0

a re a y = x, y = ln (x), x = 1, x = 2

\frac{d}{d x} (\frac{1}{1 + 3 x})