Question

Obtenha o polinomio de Taylor de ordem 4 da função f(x) = 4 x ao cubo - 2x + 3 e que f(2) = 10 . Obtenha f(x)

Answer 1

f(x) = cos(x) .....f(0) = 1
f'(x) = - sen(x)....f'(0) = 0
f''(x) = -cos(x) ....f''(0) = -1
f'''(x) = sen(x) ....f'''(0) = 0
f'''(x) = cos(x) .....f''''(0) = 1
...

$P_4= f(0)+f'(0)(x)+ \frac{f''(0)}{2!} (x-0)^2+ \frac{f'''(0)}{3!} (x-0)^3+ \frac{f''''(0)}{4!} (x-0)^4\\\\\\P_4(x)=1+0+ \frac{-1}{2*1}x^2+0+ \frac{1}{4*3*2*1}x^4 \\\\.\\\\\\\boxed{\boxed{P_4(x)=1- \frac{x^2}{2}+ \frac{x^4}{24} }}$