Numerical

\begin{tabular}{|c||c|c|c|c|}
\hline
[tex]$t$[/tex] & 1 & 2 & 5 & 7 \\
\hline
[tex]$h(t)$[/tex] & -2 & 3 & 0 & -4 \\
\hline
\end{tabular}

Selected values of the function [tex]$h$[/tex] are shown in the table above. What is the average rate of change of [tex]$h$[/tex] over the interval [tex]$[2,7]$[/tex]?



Answer :

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].