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Find the average rate of change of [tex]f(x) = 2\left(3^x\right)[/tex] over the interval [tex][-1, 2][/tex].

[tex]
\begin{array}{l}
a = -1 \\
b = 2
\end{array}
\]


Sagot :

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].