Question
Find the average rate of change of the function \( f(x)=x^{2}+5 x \) from \( x_{1}=2 \) to \( x_{2}=6 \). The average rate of change is \( \square \). (Simplify your answer.)
Ask by Herrera Cox. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The average rate of change is 13.
Solution
1. The average rate of change of a function \( f(x) \) between \( x = a \) and \( x = b \) is given by
\[
\frac{f(b) - f(a)}{b - a}.
\]
2. For \( f(x) = x^2 + 5x \), let's evaluate \( f(2) \) and \( f(6) \):
\[
f(2) = 2^2 + 5(2) = 4 + 10 = 14,
\]
\[
f(6) = 6^2 + 5(6) = 36 + 30 = 66.
\]
3. Substitute these into the formula with \( a = 2 \) and \( b = 6 \):
\[
\text{Average rate of change} = \frac{66 - 14}{6 - 2} = \frac{52}{4} = 13.
\]
The average rate of change is \( 13 \).
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Simplify this solution The Deep Dive
To find the average rate of change of the function \( f(x) = x^2 + 5x \) from \( x_1 = 2 \) to \( x_2 = 6 \), we can use the formula: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \] First, we need to calculate \( f(2) \) and \( f(6) \): \[ f(2) = 2^2 + 5 \cdot 2 = 4 + 10 = 14 \] \[ f(6) = 6^2 + 5 \cdot 6 = 36 + 30 = 66 \] Now, we can plug these values into the formula: \[ \text{Average Rate of Change} = \frac{66 - 14}{6 - 2} = \frac{52}{4} = 13 \] Thus, the average rate of change is \( 13 \).
