To solve the problem, we need to compute the average rate of change of the function $f(x) = 6x^3 + 6$ over the interval $[3,5]$. The formula for the average rate of change is:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

where $a = 3$ and $b = 5$.

Step 1: Compute $f(3)$ and $f(5)$

The function is $f(x) = 6x^3 + 6$.

• $f(3) = 6(3)^3 + 6 = 6(27) + 6 = 162 + 6 = 168$
• $f(5) = 6(5)^3 + 6 = 6(125) + 6 = 750 + 6 = 756$

Step 2: Plug values into the formula

$\text{Average Rate of Change} = \frac{f(5) - f(3)}{5 - 3}$

$\text{Average Rate of Change} = \frac{756 - 168}{2} = \frac{588}{2} = 294$

Final Answer:

The average rate of change is 294.

Would you like a detailed explanation of any step?
Here are 5 related questions to extend your understanding:

1. How is the average rate of change related to the slope of a secant line?
2. What is the difference between the average rate of change and instantaneous rate of change?
3. How do you interpret the result $294$ in this context?
4. How would the calculation change if the interval were different, say $[4,6]$?
5. What does the cubic function $6x^3 + 6$ represent geometrically?

Tip:

For polynomial functions, always substitute carefully to avoid calculation errors when finding values like $f(a)$ and $f(b)$.