This table shows values that represent an exponential function.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 7 \\
\hline
2 & 9 \\
\hline
3 & 13 \\
\hline
4 & 21 \\
\hline
5 & 37 \\
\hline
\end{tabular}

What is the average rate of change for this function for the interval from [tex]$x=3$[/tex] to [tex]$x=5$[/tex]?

A. 16
B. 8
C. 12
D. 6



Answer :

To find the average rate of change for the exponential function in the interval from [tex]\( x = 3 \)[/tex] to [tex]\( x = 5 \)[/tex], follow these steps:

1. Identify the values of [tex]\( y \)[/tex] at the given [tex]\( x \)[/tex]-values from the table.
- For [tex]\( x = 3 \)[/tex], the corresponding [tex]\( y \)[/tex]-value is [tex]\( y_3 = 13 \)[/tex].
- For [tex]\( x = 5 \)[/tex], the corresponding [tex]\( y \)[/tex]-value is [tex]\( y_5 = 37 \)[/tex].

2. Use the formula for the average rate of change, which is similar to finding the slope of the line connecting the two points [tex]\((x_3, y_3)\)[/tex] and [tex]\((x_5, y_5)\)[/tex]:
[tex]\[ \text{Average rate of change} = \frac{y_5 - y_3}{x_5 - x_3} \][/tex]

3. Substitute the values into the formula:
[tex]\[ \text{Average rate of change} = \frac{37 - 13}{5 - 3} \][/tex]

4. Perform the arithmetic operations:
[tex]\[ \text{Average rate of change} = \frac{24}{2} = 12 \][/tex]

Therefore, the average rate of change for the function in the interval from [tex]\( x = 3 \)[/tex] to [tex]\( x = 5 \)[/tex] is 12, which corresponds to option C.

Answer: C [tex]\( 12 \)[/tex]