Question

given the function h(x) = x² + x - 7, determine the average rate of change of the function over the interval -5 ≤ x ≤ 6.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( h(x) \) over the interval \([a, b]\) is given by \(\frac{h(b) - h(a)}{b - a}\). Here, \( a=-5 \) and \( b = 6 \).

Step2: Calculate \( h(-5) \)

Substitute \( x=-5 \) into \( h(x)=x^{2}+x - 7 \):
\[

$$\begin{align*} h(-5)&=(-5)^{2}+(-5)-7\\ &=25 - 5-7\\ &=13 \end{align*}$$

\]

Step3: Calculate \( h(6) \)

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

$$\begin{align*} h(6)&=6^{2}+6-7\\ &=36 + 6-7\\ &=35 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{h(b)-h(a)}{b - a}\) with \( a=-5 \), \( b = 6 \), \( h(-5)=13 \) and \( h(6)=35 \):
\[

$$\begin{align*} \frac{h(6)-h(-5)}{6-(-5)}&=\frac{35 - 13}{6 + 5}\\ &=\frac{22}{11}\\ &=2 \end{align*}$$

\]

Answer:

\( 2 \)