find the average rate of change of y=e^x between the interval of x=-3 and x=-1

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.

Learn how to calculate the average rate of change of the function y = e^x over the interval from x = -3 to x = -1 using the natural exponential function. This problem involves evaluating the function at specific points and applying the average rate of change formula. Suitable for high school students studying calculus and exponential functions.

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