Question

find the average rate of change of the function $f(x) = -1x^2 - 6x - 8$, on the interval $2,3$.

average rate of change =
give exact answer!

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). Here, \( a = 2 \), \( b = 3 \), and \( f(x)=-x^{2}-6x - 8 \).

Step2: Calculate \( f(3) \)

Substitute \( x = 3 \) into \( f(x) \):
\[

$$\begin{align*} f(3)&=- (3)^{2}-6(3)-8\\ &=-9 - 18 - 8\\ &=-35 \end{align*}$$

\]

Step3: Calculate \( f(2) \)

Substitute \( x = 2 \) into \( f(x) \):
\[

$$\begin{align*} f(2)&=- (2)^{2}-6(2)-8\\ &=-4 - 12 - 8\\ &=-24 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\) with \( a = 2 \), \( b = 3 \), \( f(3)=-35 \), and \( f(2)=-24 \):
\[

$$\begin{align*} \frac{f(3)-f(2)}{3 - 2}&=\frac{-35-(-24)}{1}\\ &=\frac{-35 + 24}{1}\\ &=-11 \end{align*}$$

\]

Answer:

\(-11\)