Rose Hills
01/20/2023 · High School
What is the average rate of change of \( h(x)=2^{x+1} \) over the interval \( [2,4] \) ?
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Step-by-step Solution
To find the average rate of change of the function \( h(x) = 2^{x+1} \) over the interval \( [2, 4] \), we use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a}
\]
where \( a = 2 \) and \( b = 4 \).
First, we need to calculate \( h(2) \) and \( h(4) \).
1. Calculate \( h(2) \):
\[
h(2) = 2^{2+1} = 2^3 = 8
\]
2. Calculate \( h(4) \):
\[
h(4) = 2^{4+1} = 2^5 = 32
\]
Now, we can substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{h(4) - h(2)}{4 - 2} = \frac{32 - 8}{4 - 2} = \frac{24}{2} = 12
\]
Thus, the average rate of change of \( h(x) = 2^{x+1} \) over the interval \( [2, 4] \) is
\[
\boxed{12}
\]
Quick Answer
The average rate of change is 12.
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