Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval [tex]$3 \leq x \leq 4$[/tex].

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
2 & 6 \\
\hline
3 & 4 \\
\hline
4 & 4 \\
\hline
5 & 6 \\
\hline
6 & 10 \\
\hline
\end{tabular}



Answer :

To find the average rate of change of the function over the interval [tex]\(3 \leq x \leq 4\)[/tex] using the given table, follow these steps:

1. Identify the relevant data points: We need the values of the function at [tex]\(x = 3\)[/tex] and [tex]\(x = 4\)[/tex].
- According to the table:
- At [tex]\(x = 3\)[/tex], [tex]\(f(3) = 4\)[/tex]
- At [tex]\(x = 4\)[/tex], [tex]\(f(4) = 4\)[/tex]

2. Use the average rate of change formula: The formula for the average rate of change of a function [tex]\( f(x) \)[/tex] over an interval [tex]\( [a, b] \)[/tex] is:
[tex]\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \][/tex]
In this case, [tex]\(a = 3\)[/tex] and [tex]\(b = 4\)[/tex].

3. Substitute the values into the formula and compute:
[tex]\[ \text{Average rate of change} = \frac{f(4) - f(3)}{4 - 3} \][/tex]
Substituting the function values, we get:
[tex]\[ \text{Average rate of change} = \frac{4 - 4}{4 - 3} = \frac{0}{1} = 0 \][/tex]

So, the average rate of change of the function over the interval [tex]\(3 \leq x \leq 4\)[/tex] is [tex]\(0\)[/tex].