Young May
09/16/2024 · High School
Cuá es la razón de cambio promedio de e(x) \( =\frac{2}{x+3} \) en el intervalo \( 1 \leq x \leq 3 \) ?
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Tutor-Verified Answer
Step-by-step Solution
Para encontrar la razón de cambio promedio de la función \( e(x) = \frac{2}{x+3} \) en el intervalo \( 1 \leq x \leq 3 \), utilizamos la fórmula de la razón de cambio promedio:
\[
\text{Razón de cambio promedio} = \frac{e(b) - e(a)}{b - a}
\]
donde \( a = 1 \) y \( b = 3 \).
Primero, calculamos \( e(1) \) y \( e(3) \):
\[
e(1) = \frac{2}{1 + 3} = \frac{2}{4} = \frac{1}{2}
\]
\[
e(3) = \frac{2}{3 + 3} = \frac{2}{6} = \frac{1}{3}
\]
Ahora, sustituimos estos valores en la fórmula de la razón de cambio promedio:
\[
\text{Razón de cambio promedio} = \frac{e(3) - e(1)}{3 - 1} = \frac{\frac{1}{3} - \frac{1}{2}}{2}
\]
Para restar \( \frac{1}{3} \) y \( \frac{1}{2} \), encontramos un denominador común, que es 6:
\[
\frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6}
\]
Entonces,
\[
\frac{1}{3} - \frac{1}{2} = \frac{2}{6} - \frac{3}{6} = -\frac{1}{6}
\]
Sustituyendo esto en la fórmula de la razón de cambio promedio:
\[
\text{Razón de cambio promedio} = \frac{-\frac{1}{6}}{2} = -\frac{1}{12}
\]
Por lo tanto, la razón de cambio promedio de \( e(x) \) en el intervalo \( 1 \leq x \leq 3 \) es:
\[
\boxed{-\frac{1}{12}}
\]
Quick Answer
La razón de cambio promedio de \( e(x) = \frac{2}{x+3} \) en el intervalo \( 1 \leq x \leq 3 \) es \( -\frac{1}{12} \).
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