Carter Mann
02/16/2023 · Junior High School
Find the average rate of change of the function \( f(x)=x^{2}+5 x \) from \( x_{1}=6 \) to \( x_{2}=7 \). The average rate of change is \( \square \). (Simplify your answer.)
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Step-by-step Solution
To find the average rate of change of the function \( f(x) = x^2 + 5x \) from \( x_1 = 6 \) to \( x_2 = 7 \), we use the formula for the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
First, we need to calculate \( f(6) \) and \( f(7) \).
Calculating \( f(6) \):
\[
f(6) = 6^2 + 5 \cdot 6 = 36 + 30 = 66
\]
Calculating \( f(7) \):
\[
f(7) = 7^2 + 5 \cdot 7 = 49 + 35 = 84
\]
Now we can substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{f(7) - f(6)}{7 - 6} = \frac{84 - 66}{7 - 6} = \frac{18}{1} = 18
\]
Thus, the average rate of change of the function from \( x_1 = 6 \) to \( x_2 = 7 \) is
\[
\boxed{18}
\]
Quick Answer
The average rate of change is 18.
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