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

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

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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 $$ 48x + 24\Delta x $$.

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

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

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