To calculate the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\([0, 4]\)[/tex], follow these steps:
1. Identify the values of [tex]\(f(x)\)[/tex] at [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]:
- According to the given table,
- When [tex]\(x = 0\)[/tex], [tex]\(f(0) = 16\)[/tex]
- When [tex]\(x = 4\)[/tex], [tex]\(f(4) = 256\)[/tex]
2. Calculate the change in [tex]\(f(x)\)[/tex]:
- The change in [tex]\(f(x)\)[/tex] over the interval from [tex]\(x = 0\)[/tex] to [tex]\(x = 4\)[/tex] is:
[tex]\[
f(4) - f(0) = 256 - 16 = 240
\][/tex]
3. Calculate the change in [tex]\(x\)[/tex]:
- The change in [tex]\(x\)[/tex] over the interval from [tex]\(x = 0\)[/tex] to [tex]\(x = 4\)[/tex] is:
[tex]\[
4 - 0 = 4
\][/tex]
4. Calculate the average rate of change:
- The average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\([0, 4]\)[/tex] is given by the formula:
[tex]\[
\text{Average rate of change} = \frac{\text{Change in } f(x)}{\text{Change in } x}
\][/tex]
Substituting the values, we get:
[tex]\[
\text{Average rate of change} = \frac{240}{4} = 60.0
\][/tex]
Therefore, the average rate of change of [tex]\( f(x) \)[/tex] over the interval [tex]\([0, 4]\)[/tex] is [tex]\(\boxed{60.0}\)[/tex].
