given the function f(x)=-x² - x - 12, find th...
given the function f(x)=-x² - x - 12, find the difference quotient (f(x + h)-f(x))/h. (f(x + h)-f(x))/h =
Answer
# Explanation:
## Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)$:
\[
\begin{align*}
f(x + h)&=-(x + h)^2-(x + h)-12\\
&=-(x^{2}+2xh+h^{2})-x - h-12\\
&=-x^{2}-2xh - h^{2}-x - h-12
\end{align*}
\]
## Step2: Calculate $f(x + h)-f(x)$
\[
\begin{align*}
f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}-x - h-12)-(-x^{2}-x - 12)\\
&=-x^{2}-2xh - h^{2}-x - h-12 + x^{2}+x + 12\\
&=-2xh - h^{2}-h
\end{align*}
\]
## Step3: Find the difference - quotient
\[
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{-2xh - h^{2}-h}{h}\\
&=\frac{h(-2x - h - 1)}{h}\\
&=-2x - h - 1
\end{align*}
\]
# Answer:
$-2x - h - 1$