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]
