given the function $f(x) = x^2 - 3x - 5$, determine the average rate of change of the function over the interval $-2 \leq x \leq 3$.

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $ f(x) $ over the interval $[a, b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $ a=-2 $ and $ b = 3 $.

## Step2: Calculate $ f(-2) $
Substitute $ x=-2 $ into $ f(x)=x^{2}-3x - 5 $:
$$
\begin{align*}
f(-2)&=(-2)^{2}-3(-2)-5\
&=4 + 6-5\
&=5
\end{align*}
$$

## Step3: Calculate $ f(3) $
Substitute $ x = 3 $ into $ f(x)=x^{2}-3x - 5 $:
$$
\begin{align*}
f(3)&=(3)^{2}-3(3)-5\
&=9-9 - 5\
&=-5
\end{align*}
$$

## Step4: Calculate the average rate of change
Using the formula $\frac{f(b)-f(a)}{b - a}$ with $ a=-2 $, $ b = 3 $, $ f(-2)=5 $ and $ f(3)=-5 $:
$$
\begin{align*}
\frac{f(3)-f(-2)}{3-(-2)}&=\frac{-5 - 5}{3 + 2}\
&=\frac{-10}{5}\
&=-2
\end{align*}
$$

# Answer:
$-2$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Calculate \( f(-2) \)

Step3: Calculate \( f(3) \)

Step4: Calculate the average rate of change

Answer: