To find the average rate of change of the function [tex]\( f(x) = 2(3^x) \)[/tex] over the interval [tex]\([-1, 2]\)[/tex], follow these steps:
1. Determine the function values at the boundaries of the interval.
Evaluate [tex]\( f(x) \)[/tex] at the endpoints [tex]\( x = -1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
f(a) = f(-1) = 2 \cdot 3^{-1} = 2 \cdot \frac{1}{3} = \frac{2}{3}
\][/tex]
[tex]\[
f(b) = f(2) = 2 \cdot 3^2 = 2 \cdot 9 = 18
\][/tex]
2. Calculate the difference in the function values.
Find the difference:
[tex]\[
f(b) - f(a) = 18 - \frac{2}{3}
\][/tex]
To simplify the calculation, convert [tex]\(\frac{2}{3}\)[/tex] to a decimal form:
[tex]\[
\frac{2}{3} \approx 0.6667
\][/tex]
So,
[tex]\[
18 - 0.6667 = 17.3333
\][/tex]
3. Determine the length of the interval.
Compute [tex]\( b - a \)[/tex]:
[tex]\[
b - a = 2 - (-1) = 2 + 1 = 3
\][/tex]
4. Calculate the average rate of change.
Use the formula for the average rate of change:
[tex]\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a} = \frac{17.3333}{3} \approx 5.7778
\][/tex]
Therefore, the average rate of change of the function [tex]\( f(x) = 2(3^x) \)[/tex] over the interval [tex]\([-1, 2]\)[/tex] is approximately [tex]\( 5.7778 \)[/tex].
To summarize:
- [tex]\( f(-1) = \frac{2}{3} \approx 0.6667 \)[/tex]
- [tex]\( f(2) = 18 \)[/tex]
- The average rate of change is approximately [tex]\( 5.7778 \)[/tex].
