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

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

Where $a = -1$ and $b = 7$.

First, calculate the function values at $x = -1$ and $x = 7$:

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

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

Now, apply the average rate of change formula:

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

So, the average rate of change of $f(x)$ from $x = -1$ to $x = 7$ is:

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

Would you like me to simplify this further or provide a numerical approximation?

---

1. What is the significance of the average rate of change in calculus?
2. How does the exponential function affect the rate of change?
3. What would the average rate of change be for different intervals?
4. How does the sign of the average rate of change relate to the function's behavior?
5. Can you apply this concept to other functions, such as polynomials or logarithms?

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