To determine the average rate of change of $f$ over the interval $[1, 5]$, we can use the formula for the average rate of change:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

where:

• $a = 1$
• $b = 5$
• $f(1) = 2$
• $f(5) = 14$

Plugging in the values:

$\frac{f(5) - f(1)}{5 - 1} = \frac{14 - 2}{5 - 1} = \frac{12}{4} = 3$

Thus, the average rate of change of $f$ over the interval $[1, 5]$ is $3$.

Answer:

The correct choice is C: $\frac{14 - 2}{5 - 1}$.

Would you like more details or have any questions?

Related Questions:

1. What is the average rate of change of $f$ over a different interval, such as $[2, 4]$?
2. How would the result change if $f(5)$ were a different value?
3. How can the average rate of change be interpreted graphically?
4. What is the significance of the interval endpoints $[1, 5]$ in this context?
5. How does average rate of change differ from instantaneous rate of change?

Tip:

When finding the average rate of change, remember it essentially measures the "slope" between two points on the function's graph over a specified interval.