Lambert Grant
07/26/2023 · Middle School
¿Cuál es la razón de cambio promedio de \( f(x)=\sqrt{x^{3}-4} \) en el intervalo \( 2 \leq x \leq 5 \) ?
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Step-by-step Solution
La razón de cambio promedio de una función \( f(x) \) en un intervalo \( [a, b] \) se calcula utilizando la fórmula:
\[
\text{Razón de cambio promedio} = \frac{f(b) - f(a)}{b - a}
\]
En este caso, tenemos la función \( f(x) = \sqrt{x^3 - 4} \) y el intervalo \( [2, 5] \). Por lo tanto, \( a = 2 \) y \( b = 5 \).
Primero, calculamos \( f(2) \) y \( f(5) \):
1. Calcular \( f(2) \):
\[
f(2) = \sqrt{2^3 - 4} = \sqrt{8 - 4} = \sqrt{4} = 2
\]
2. Calcular \( f(5) \):
\[
f(5) = \sqrt{5^3 - 4} = \sqrt{125 - 4} = \sqrt{121} = 11
\]
Ahora, sustituimos estos valores en la fórmula de la razón de cambio promedio:
\[
\text{Razón de cambio promedio} = \frac{f(5) - f(2)}{5 - 2} = \frac{11 - 2}{5 - 2} = \frac{9}{3} = 3
\]
Por lo tanto, la razón de cambio promedio de \( f(x) \) en el intervalo \( [2, 5] \) es \( \boxed{3} \).
Quick Answer
La razón de cambio promedio de \( f(x)=\sqrt{x^{3}-4} \) en el intervalo \( 2 \leq x \leq 5 \) es 3.
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