3. find the average rate of change of ( f(x)=-2 x^{3}+15 x^{2}-17 ) over the interval ( 6,8 ).
2. write your answer as an integer, fraction, or decimal rounded to the nearest tenth.
simplify any fractions.

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a = 6$, $b=8$, and $f(x)=-2x^{3}+15x^{2}-17$.
## Step2: Calculate $f(8)$
Substitute $x = 8$ into $f(x)$:
$$
\begin{align*}
f(8)&=-2\times(8)^{3}+15\times(8)^{2}-17\
&=-2\times512 + 15\times64-17\
&=-1024+960 - 17\
&=-81
\end{align*}
$$
## Step3: Calculate $f(6)$
Substitute $x = 6$ into $f(x)$:
$$
\begin{align*}
f(6)&=-2\times(6)^{3}+15\times(6)^{2}-17\
&=-2\times216+15\times36 - 17\
&=-432 + 540-17\
&=91
\end{align*}
$$
## Step4: Compute the average rate of change
Using the formula $\frac{f(b)-f(a)}{b - a}$, with $a = 6$, $b = 8$, $f(8)=-81$, and $f(6)=91$:
$$
\begin{align*}
\frac{f(8)-f(6)}{8 - 6}&=\frac{-81-91}{2}\
&=\frac{-172}{2}\
&=- 86
\end{align*}
$$

# Answer:
$-86$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Calculate \(f(8)\)

Step3: Calculate \(f(6)\)

Step4: Compute the average rate of change

Answer: