find the average rate of change for y= e^x between x=2 and x=6

To find the average rate of change of the function \( y = e^x \) between \( x = 2 \) and \( x = 6 \), we use the formula for the average rate of change of a function over an interval \([a, b]\):

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

Here, \( f(x) = e^x \), \( a = 2 \), and \( b = 6 \).

First, we calculate \( f(2) \) and \( f(6) \):
\[
f(2) = e^2
\]
\[
f(6) = e^6
\]

Next, we plug these values into the formula:

\[
\text{Average Rate of Change} = \frac{e^6 - e^2}{6 - 2}
\]

Simplify the denominator:

\[
\text{Average Rate of Change} = \frac{e^6 - e^2}{4}
\]

To get a numerical value, we can approximate \( e \approx 2.718 \):

\[
e^2 \approx 7.389
\]
\[
e^6 \approx 403.429
\]

So,

\[
\text{Average Rate of Change} \approx \frac{403.429 - 7.389}{4} = \frac{396.040}{

Learn how to calculate the average rate of change for the function y = e^x between x = 2 and x = 6. This problem involves using the Average Rate of Change formula with exponential functions, providing step-by-step calculations. Suitable for advanced high school students.

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