Question

given the function $f(x) = -x^2 - 8x + 21$, determine the average rate of change of the function over the interval $-10 \leq x \leq 0$.

Explanation:

Step1: Recall the average rate of change formula

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, \( a=-10 \) and \( b = 0 \).

Step2: Calculate \( f(-10) \)

Substitute \( x=-10 \) into \( f(x)=-x^{2}-8x + 21 \):
\[

$$\begin{align*} f(-10)&=-(-10)^{2}-8(-10)+21\\ &=-100 + 80+21\\ &=-19 \end{align*}$$

\]

Step3: Calculate \( f(0) \)

Substitute \( x = 0 \) into \( f(x)=-x^{2}-8x + 21 \):
\[
f(0)=-(0)^{2}-8(0)+21=21
\]

Step4: Apply the average rate of change formula

Now, use \( a=-10 \), \( b = 0 \), \( f(-10)=-19 \) and \( f(0)=21 \) in the formula \(\frac{f(b)-f(a)}{b - a}\):
\[

$$\begin{align*} \frac{f(0)-f(-10)}{0-(-10)}&=\frac{21-(-19)}{0 + 10}\\ &=\frac{21 + 19}{10}\\ &=\frac{40}{10}\\ &=4 \end{align*}$$

\]

Answer:

\( 4 \)