Find the average rate of change for the function f(x)=3e^-x from x=-1 to =7

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?

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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.

Learn how to calculate the average rate of change for the function f(x) = 3e^-x from x = -1 to x = 7. Understand the formula, steps, and numerical approximation involved in finding this rate of change. Suitable for high school students and anyone studying calculus and exponential functions.

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