given the function f(x)=9x² - 1/2x, find the ...
given the function f(x)=9x² - 1/2x, find the difference quotient (f(-3 + h)-f(-3))/h. enter exact answers only, no approximations. (f(-3 + h)-f(-3))/h
Answer
# Explanation:
## Step1: Find $f(-3 + h)$
Substitute $x=-3 + h$ into $f(x)=9x^{2}-\frac{1}{2}x$.
\[
\begin{align*}
f(-3 + h)&=9(-3 + h)^{2}-\frac{1}{2}(-3 + h)\\
&=9(9 - 6h+h^{2})+\frac{3}{2}-\frac{1}{2}h\\
&=81-54h + 9h^{2}+\frac{3}{2}-\frac{1}{2}h\\
&=9h^{2}-54h-\frac{1}{2}h + 81+\frac{3}{2}\\
&=9h^{2}-\frac{109}{2}h+\frac{162 + 3}{2}\\
&=9h^{2}-\frac{109}{2}h+\frac{165}{2}
\end{align*}
\]
## Step2: Find $f(-3)$
Substitute $x = - 3$ into $f(x)=9x^{2}-\frac{1}{2}x$.
\[
\begin{align*}
f(-3)&=9(-3)^{2}-\frac{1}{2}(-3)\\
&=9\times9+\frac{3}{2}\\
&=81+\frac{3}{2}\\
&=\frac{162 + 3}{2}\\
&=\frac{165}{2}
\end{align*}
\]
## Step3: Calculate $f(-3 + h)-f(-3)$
\[
\begin{align*}
f(-3 + h)-f(-3)&=(9h^{2}-\frac{109}{2}h+\frac{165}{2})-\frac{165}{2}\\
&=9h^{2}-\frac{109}{2}h
\end{align*}
\]
## Step4: Calculate the difference - quotient
\[
\begin{align*}
\frac{f(-3 + h)-f(-3)}{h}&=\frac{9h^{2}-\frac{109}{2}h}{h}\\
&=\frac{h(9h-\frac{109}{2})}{h}\\
&=9h-\frac{109}{2}
\end{align*}
\]
# Answer:
$9h-\frac{109}{2}$