To determine the average rate of change of the function [tex]\( h \)[/tex] over the interval [tex]\([2, 7]\)[/tex], we follow these steps:
1. Identify the values from the table:
[tex]\[
\begin{array}{|c||c|c|c|c|}
\hline
t & 1 & 2 & 5 & 7 \\
\hline
h(t) & -2 & 3 & 0 & -4 \\
\hline
\end{array}
\][/tex]
From this table, we see that when [tex]\( t = 2 \)[/tex], [tex]\( h(t) = 3 \)[/tex], and when [tex]\( t = 7 \)[/tex], [tex]\( h(t) = -4 \)[/tex].
2. Recall the formula for the average rate of change:
[tex]\[
\text{Average Rate of Change} = \frac{h(t_2) - h(t_1)}{t_2 - t_1}
\][/tex]
Here, [tex]\( t_1 = 2 \)[/tex], [tex]\( t_2 = 7 \)[/tex], [tex]\( h(t_1) = 3 \)[/tex], and [tex]\( h(t_2) = -4 \)[/tex].
3. Substitute the given values into the formula:
[tex]\[
\text{Average Rate of Change} = \frac{h(7) - h(2)}{7 - 2}
\][/tex]
4. Calculate the change in [tex]\( h \)[/tex] values:
[tex]\[
h(7) - h(2) = -4 - 3 = -7
\][/tex]
5. Calculate the change in [tex]\( t \)[/tex] values:
[tex]\[
7 - 2 = 5
\][/tex]
6. Divide the change in [tex]\( h \)[/tex] by the change in [tex]\( t \)[/tex]:
[tex]\[
\text{Average Rate of Change} = \frac{-7}{5} = -1.4
\][/tex]
So, the average rate of change of [tex]\( h \)[/tex] over the interval [tex]\([2, 7]\)[/tex] is [tex]\(\boxed{-1.4}\)[/tex].
