To find the approximate average rate of change of the function $f(x) = 50(1.021)^x$ over the interval from $x = 7$ to $x = 10$, we can use the following formula for the average rate of change:

$\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Here, $x_1 = 7$, $x_2 = 10$, and the function is given as $f(x) = 50(1.021)^x$. Let's calculate the values of $f(7)$ and $f(10)$ and then find the average rate of change.

I'll calculate this now.The values of the function are:

• $f(7) = 57.83$
• $f(10) = 61.55$

The approximate average rate of change over the interval from $x = 7$ to $x = 10$ is:

$\frac{61.55 - 57.83}{10 - 7} = 1.24$

So, the average rate of change is approximately $1.24$.

Do you want any additional details or have any questions?

Here are 5 related questions:

1. How does the rate of change differ for different types of functions (e.g., linear vs exponential)?
2. What is the significance of the base 1.021 in the exponential function?
3. How would the average rate of change differ if the interval were $x = 0$ to $x = 7$?
4. Can you calculate the instantaneous rate of change at a specific point on this function?
5. How does the rate of change of exponential functions compare to polynomial functions over large intervals?

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