Pregunta
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
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The average rate of change is 13.
Solución
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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Simplifique esta solución 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 \( 13 \).
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