To find the average rate of change of the function $y = e^x$ over the interval $[x_1, x_2]$, you can use the formula:

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

Given the interval $[-3, -1]$, we have:

• $x_1 = -3$
• $x_2 = -1$

The function is $y = e^x$, so we need to find $f(-3)$ and $f(-1)$:

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

Now, plug these values into the formula:

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

Simplify the denominator:

$\text{Average Rate of Change} = \frac{e^{-1} - e^{-3}}{-1 + 3} = \frac{e^{-1} - e^{-3}}{2}$

Thus, the average rate of change is:

$\text{Average Rate of Change} = \frac{e^{-1} - e^{-3}}{2}$

We can also approximate the numerical value:

$e^{-1} = \frac{1}{e} \approx 0.3679$ $e^{-3} = \frac{1}{e^3} \approx 0.0498$

So,

$\text{Average Rate of Change} \approx \frac{0.3679 - 0.0498}{2} = \frac{0.3181}{2} \approx 0.1591$

Therefore, the average rate of change of $y = e^x$ between $x = -3$ and $x = -1$ is approximately $0.1591$.

Would you like further details or have any questions about this calculation?

Here are 5 numbered questions for you to ask:

1. How do you find the average rate of change for a linear function?
2. Can you explain the concept of exponential growth?
3. How does the derivative of $e^x$ relate to its rate of change?
4. What are some real-life applications of exponential functions?
5. How do you differentiate $e^{ax}$ where $a$ is a constant?

Tip: When dealing with exponential functions, remember that the base $e$ is a constant approximately equal to 2.71828, and it plays a crucial role in calculus due to its unique properties related to differentiation and integration.