The question in the image asks for the average rate of change of the function $f(x) = 9x^2 - 5$ on the interval $[2, t]$, where the answer should involve the variable $t$.

Solution:

The average rate of change of a function over an interval $[a, b]$ is given by the formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$ In this case, $f(x) = 9x^2 - 5$, $a = 2$, and $b = t$.

1. Step 1: Calculate $f(t)$ and $f(2)$.

$f(t) = 9t^2 - 5$ $f(2) = 9(2)^2 - 5 = 36 - 5 = 31$

2. Step 2: Apply the average rate of change formula.

$\text{Average Rate of Change} = \frac{f(t) - f(2)}{t - 2} = \frac{(9t^2 - 5) - 31}{t - 2} = \frac{9t^2 - 36}{t - 2}$

So, the average rate of change is: $\boxed{\frac{9t^2 - 36}{t - 2}}$

This is the expression for the average rate of change involving $t$.

Do you need more details or further explanation on any part of this solution?

Related Questions:

1. How is the average rate of change different from the instantaneous rate of change?
2. Can you derive the general formula for the average rate of change for a polynomial function?
3. What is the significance of the result when $t = 2$ in this problem?
4. How would you interpret the average rate of change geometrically?
5. Can you calculate the derivative of the function $f(x) = 9x^2 - 5$?

Tip:

Remember, the average rate of change over an interval is the slope of the secant line that connects two points on the graph of the function over that interval.