Question

El polinomio de taylor de grado 2 en x0 (subcero)= -1 de F(x) es P(x)= -8-5x-3x^2. Sea G(x)= x.F.(-x^2+3), calcular G'(2)

Answer 1

Solución

$\displaystyle P(x)=\sum_{n=0}^{2}\dfrac{F^{(n)}(-1)}{n!}(x+1)^n\\ \\ \\ P(x)=F(-1)+F'(-1)(x+1)+\dfrac{F''(-1)}{2}(x+1)^2\\ \\ \\ \texttt{Seg\'un los datos tenemos: }\\ \\ ~~~~~~~~~~F(-1)=-9~,~F'(-1)=7~,~F''(-1)=-12\\ \\ \texttt{Por otra parte:}\\ \\ G(x)=x\cdot{F(-x^2+3)}\\ \\ G'(x)=F(-x^2+3)+x\cdot F'(-x^2+3)\cdot(-2x)\\ \\ G'(x)=F(-x^2+3)-2x^2\cdot F'(-x^2+3)\\ \\ G'(2)=F(-1)-8F'(-1)\\ \\ G'(2)=-9-8(7)\\ \\ \boxed{~\boxed{G'(2)=-65}~}$