To find the average rate of change of the function $f(x) = 3e^{-x}$ from $x = -1$ to $x = 7$, we use the following formula:

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

Here, $a = -1$ and $b = 7$.

1. Calculate $f(-1)$:

$f(-1) = 3e^{-(-1)} = 3e^{1} = 3e$

2. Calculate $f(7)$:

$f(7) = 3e^{-7}$

3. Apply the values to the formula:

$\text{Average Rate of Change} = \frac{3e^{-7} - 3e}{7 - (-1)} = \frac{3e^{-7} - 3e}{8}$

4. Simplify:

$\text{Average Rate of Change} = \frac{3(e^{-7} - e)}{8}$

Thus, the average rate of change of the function $f(x) = 3e^{-x}$ from $x = -1$ to $x = 7$ is $\frac{3(e^{-7} - e)}{8}$.

Would you like to see the numerical approximation or have any other questions?

---

Here are 5 related questions:

1. How does the average rate of change compare with the instantaneous rate of change at a point?
2. What is the significance of the base of the natural exponential function $e$?
3. How would the average rate of change differ if the interval was from $x = 0$ to $x = 7$?
4. Can you find the derivative of the function $f(x) = 3e^{-x}$?
5. How does the function $3e^{-x}$ behave as $x$ approaches infinity?

Tip: The average rate of change over an interval gives the slope of the secant line connecting two points on the curve.