Question

given $f(x) = x^2 + 7x$, find the average rate of change of $f(x)$ on the interval $-4, -4 + h$. your answer will be an expression involving $h$.

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, \( a=-4 \) and \( b = -4 + h \).

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

Substitute \( x=-4 \) into \( f(x)=x^{2}+7x \):
\( f(-4)=(-4)^{2}+7\times(-4)=16 - 28=-12 \)

Step3: Find \( f(-4 + h) \)

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

$$\begin{align*} f(-4 + h)&=(-4 + h)^{2}+7\times(-4 + h)\\ &=16-8h+h^{2}-28 + 7h\\ &=h^{2}-h - 12 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), we have:
\[

$$\begin{align*} \frac{f(-4 + h)-f(-4)}{(-4 + h)-(-4)}&=\frac{(h^{2}-h - 12)-(-12)}{h}\\ &=\frac{h^{2}-h - 12 + 12}{h}\\ &=\frac{h^{2}-h}{h}\\ &=\frac{h(h - 1)}{h}\\ &=h - 1\quad (h eq0) \end{align*}$$

\]

Answer:

\( h - 1 \)