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Resolver los siguientes ejercicios aplicando los conocimientos previos. \( \begin{array}{l} \text { 1. Sea } f(x)=2(x+2)^{3}-2(x-2)^{3} \quad \text { Hallar la tasa de cambio promedio: } \\ \frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}\end{array} \)

Ask by Wood Brewer. in Colombia
Mar 13,2025

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La tasa de cambio promedio es \( 48x + 24\Delta x \).

Solution

Alright, I need to solve the given problem step by step. The function provided is \( f(x) = 2(x + 2)^3 - 2(x - 2)^3 \), and I need to find the average rate of change, which is given by \( \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x} \). First, let's understand what the average rate of change represents. It's essentially the slope of the secant line between two points on the graph of the function. In this case, the two points are \( (x, f(x)) \) and \( (x + \Delta x, f(x + \Delta x)) \). To find \( \frac{\Delta y}{\Delta x} \), I need to compute \( f(x + \Delta x) \) and then subtract \( f(x) \) from it, finally dividing by \( \Delta x \). Let's start by expanding \( f(x) \): \[ f(x) = 2(x + 2)^3 - 2(x - 2)^3 \] Expanding each term: \[ (x + 2)^3 = x^3 + 6x^2 + 12x + 8 \] \[ (x - 2)^3 = x^3 - 6x^2 + 12x - 8 \] Now, multiply each by 2: \[ 2(x + 2)^3 = 2x^3 + 12x^2 + 24x + 16 \] \[ 2(x - 2)^3 = 2x^3 - 12x^2 + 24x - 16 \] Subtract the second from the first: \[ f(x) = (2x^3 + 12x^2 + 24x + 16) - (2x^3 - 12x^2 + 24x - 16) = 24x^2 + 32 \] So, the simplified form of \( f(x) \) is: \[ f(x) = 24x^2 + 32 \] Now, let's find \( f(x + \Delta x) \): \[ f(x + \Delta x) = 24(x + \Delta x)^2 + 32 \] Expanding \( (x + \Delta x)^2 \): \[ (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2 \] Multiply by 24: \[ 24(x + \Delta x)^2 = 24x^2 + 48x\Delta x + 24(\Delta x)^2 \] Add 32: \[ f(x + \Delta x) = 24x^2 + 48x\Delta x + 24(\Delta x)^2 + 32 \] Now, compute \( f(x + \Delta x) - f(x) \): \[ f(x + \Delta x) - f(x) = (24x^2 + 48x\Delta x + 24(\Delta x)^2 + 32) - (24x^2 + 32) = 48x\Delta x + 24(\Delta x)^2 \] Finally, divide by \( \Delta x \) to find the average rate of change: \[ \frac{\Delta y}{\Delta x} = \frac{48x\Delta x + 24(\Delta x)^2}{\Delta x} = 48x + 24\Delta x \] So, the average rate of change is \( 48x + 24\Delta x \).

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Para encontrar la tasa de cambio promedio de la función \( f(x)=2(x+2)^{3}-2(x-2)^{3} \), primero necesitas calcular \( f(x+\Delta x) \). Sustituyes \( (x+\Delta x) \) en la función: \[ f(x+\Delta x) = 2((x+\Delta x)+2)^{3}-2((x+\Delta x)-2)^{3} \] Luego, puedes calcular la diferencia \( f(x+\Delta x) - f(x) \) y dividir por \( \Delta x \). Este procedimiento te mostrará cómo cambia la función con respecto a pequeños incrementos en \( x \). Es importante recordar que el resultado de la tasa de cambio promedio te da una idea de la inclinación de la función en el intervalo considerado. A medida que \( \Delta x \) se hace más pequeño, esta tasa de cambio promedio se asemejará a la derivada en ese punto, dándote una idea de la "pendiente instantánea" de la función en \( x \).

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