Question

given the function $f(x) = x^2 + 7x + 6$, determine the average rate of change of the function over the interval $-4 \leq x \leq -1$.

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, \( a=-4 \) and \( b = - 1\).

Step2: Calculate \( f(-4) \)

Substitute \( x=-4 \) into \( f(x)=x^{2}+7x + 6 \):
\[

$$\begin{align*} f(-4)&=(-4)^{2}+7\times(-4)+6\\ &=16-28 + 6\\ &=-6 \end{align*}$$

\]

Step3: Calculate \( f(-1) \)

Substitute \( x = - 1\) into \( f(x)=x^{2}+7x + 6 \):
\[

$$\begin{align*} f(-1)&=(-1)^{2}+7\times(-1)+6\\ &=1-7 + 6\\ &=0 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\) with \( a=-4 \), \( b=-1 \), \( f(-4)=-6 \) and \( f(-1) = 0\):
\[

$$\begin{align*} \frac{f(-1)-f(-4)}{-1-(-4)}&=\frac{0-(-6)}{-1 + 4}\\ &=\frac{6}{3}\\ &=2 \end{align*}$$

\]

Answer:

\( 2 \)