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Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval [tex]$18 \leq x \leq 24$[/tex].

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
12 & 41 \\
\hline
18 & 40 \\
\hline
24 & 39 \\
\hline
30 & 38 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
To find the average rate of change of the function over the interval [tex]\( 18 \leq x \leq 24 \)[/tex], follow these steps:

1. Identify the given points on the interval:
- Let [tex]\( x_1 = 18 \)[/tex] and [tex]\( x_2 = 24 \)[/tex].
- The corresponding function values are [tex]\( f(x_1) = f(18) = 40 \)[/tex] and [tex]\( f(x_2) = f(24) = 39 \)[/tex].

2. Use the formula for the average rate of change:
[tex]\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]

3. Substitute the values into the formula:
[tex]\[ \text{Average rate of change} = \frac{f(24) - f(18)}{24 - 18} = \frac{39 - 40}{24 - 18} \][/tex]

4. Perform the calculations:
[tex]\[ \text{Average rate of change} = \frac{39 - 40}{6} = \frac{-1}{6} \][/tex]

Therefore, the average rate of change of the function over the interval [tex]\( 18 \leq x \leq 24 \)[/tex] is:
[tex]\[ -0.16666666666666666 \][/tex]

In simplest fractional form, this is [tex]\(-\frac{1}{6}\)[/tex].

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