find the difference quotient and simplify. f(...
find the difference quotient and simplify. f(x)=-2x^2 - 4x + 2 the difference quotient of f(x) is
Answer
# Explanation:
## Step1: Definir la fórmula del cociente de diferencia
El cociente de diferencia para una función $f(x)$ es $\frac{f(x + h)-f(x)}{h}$, donde $h\neq0$.
## Step2: Encontrar $f(x + h)$
Sustituir $x$ por $x + h$ en $f(x)=-2x^{2}-4x + 2$:
\[
\begin{align*}
f(x + h)&=-2(x + h)^{2}-4(x + h)+2\\
&=-2(x^{2}+2xh+h^{2})-4x-4h + 2\\
&=-2x^{2}-4xh-2h^{2}-4x-4h + 2
\end{align*}
\]
## Step3: Calcular $f(x + h)-f(x)$
\[
\begin{align*}
f(x + h)-f(x)&=(-2x^{2}-4xh-2h^{2}-4x-4h + 2)-(-2x^{2}-4x + 2)\\
&=-2x^{2}-4xh-2h^{2}-4x-4h + 2 + 2x^{2}+4x - 2\\
&=-4xh-2h^{2}-4h
\end{align*}
\]
## Step4: Calcular el cociente de diferencia
\[
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{-4xh-2h^{2}-4h}{h}\\
&=\frac{h(-4x-2h - 4)}{h}\\
&=-4x-2h-4
\end{align*}
\]
# Answer:
$-4x-2h - 4$