Polinomio de Taylor
Ejemplo: Obtener el polinomio de Taylor de la función f( x )=cosx en x=0 y en x= π 4 .
Derivando
f(x)=cosx f(0)=1 f( π 4 )= 2 2 f'(x)=−senx f'(0)=0 f'( π 4 )=− 2 2 f''(x)=−cosx f''(0)=−1 f''( π 4 )=− 2 2 f'''(x)=senx f'''(0)=0 f'''( π 4 )= 2 2 f (iv (x)=cosx f (iv (0)=1 f (iv ( π 4 )= 2 2 ... ... ... f (2n (x)= (−1) n cosx f (2n (0)= ( −1 ) n f (2n ( π 4 )= (−1) n 2 2 f (2n+1 (x)= (−1) n+1 senx f (2n+1 (0)=0 f (2n+1 ( π 4 )= (−1) n+1 2 2 por lo que: T 2n [cosx;0]=1− x 2 2! + x 4 4! − x 6 6! +...+ (−1) n x 2n (2n)! T 2n+1 [cosx; π 4 ]= 2 2 ( 1−( x− π 4 )− 1 2 ( x− π 4 ) 2 +... ...+ (−1) n (2n)! ( x− π 4 ) 2n + (−1) n+1 (2n+1)! ( x− π 4 ) 2n+1 )