To find the average rate of change of the function $f(x) = 3x^2 + 2x - 5$ on the interval $[x, x+h]$, we use the formula:

$\text{Average Rate of Change} = \frac{f(x+h) - f(x)}{h}$

1. First, calculate $f(x+h)$. Substitute $x+h$ into the function:

$f(x+h) = 3(x+h)^2 + 2(x+h) - 5$

Expand this expression:

$f(x+h) = 3(x^2 + 2xh + h^2) + 2(x+h) - 5$ $= 3x^2 + 6xh + 3h^2 + 2x + 2h - 5$

2. Now, subtract $f(x)$ from $f(x+h)$:

$f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 + 2x + 2h - 5) - (3x^2 + 2x - 5)$ $= 6xh + 3h^2 + 2h$

3. Finally, divide the result by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 + 2h}{h}$ $= 6x + 3h + 2$

Thus, the average rate of change of $f(x)$ on the interval $[x, x+h]$ is:

$6x + 3h + 2$

Would you like more details or have any questions? Here are 5 related questions:

1. How do you interpret the average rate of change geometrically?
2. What is the significance of the term $3h$ as $h \to 0$?
3. How does this expression simplify when $h \to 0$, representing the instantaneous rate of change?
4. Can you apply the same method to find the average rate of change for a linear function?
5. What are the steps to compute the derivative of $f(x)$ using the limit definition?

Tip: The average rate of change is the slope of the secant line between two points on the function.