To calculate the average rate of change for the exponential function between $x = 1$ and $x = 7$, we use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Where:

• $x_1 = 1$
• $x_2 = 7$
• $f(x)$ is the given exponential function.

From your image, the exponential function appears to be $f(x) = 3^x$.

Step-by-Step Calculation:

1. Evaluate $f(x_1) = f(1)$: $f(1) = 3^1 = 3$

2. Evaluate $f(x_2) = f(7)$: $f(7) = 3^7 = 2187$

3. Calculate the average rate of change: $\text{Average Rate of Change} = \frac{f(7) - f(1)}{7 - 1}$ Substituting the values: $\text{Average Rate of Change} = \frac{2187 - 3}{6} = \frac{2184}{6} = 364$

Final Answer:

The average rate of change of the function $f(x) = 3^x$ between $x = 1$ and $x = 7$ is 364.

---

Would you like further details or have any questions?
Here are 5 follow-up questions you might consider:

1. How does the average rate of change differ from the instantaneous rate of change?
2. Can you generalize the average rate of change formula for other exponential functions?
3. How would the average rate of change behave as the interval between $x_1$ and $x_2$ increases?
4. What is the geometric interpretation of the average rate of change in an exponential function?
5. How does the base of the exponential function ($3$ in this case) influence the average rate of change?

Tip: The average rate of change gives a good approximation of the "speed" of growth or decay in exponential functions over a given interval.