Question

find the average rate of change for the function between the given values. 17) f(x) = 4x^3 - 8x^2 - 1; from -4 to 1

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-4$, $b = 1$, and $f(x)=4x^{3}-8x^{2}-1$.

Step2: Calculate $f(a)$

First, find $f(-4)$:
\[

$$\begin{align*} f(-4)&=4(-4)^{3}-8(-4)^{2}-1\\ &=4\times(-64)-8\times16 - 1\\ &=-256-128 - 1\\ &=-385 \end{align*}$$

\]

Step3: Calculate $f(b)$

Then, find $f(1)$:
\[

$$\begin{align*} f(1)&=4\times(1)^{3}-8\times(1)^{2}-1\\ &=4 - 8-1\\ &=-5 \end{align*}$$

\]

Step4: Calculate the average rate of change

Now, use the formula $\frac{f(b)-f(a)}{b - a}$:
\[

$$\begin{align*} \frac{f(1)-f(-4)}{1-(-4)}&=\frac{-5-(-385)}{1 + 4}\\ &=\frac{-5 + 385}{5}\\ &=\frac{380}{5}\\ &=76 \end{align*}$$

\]

Answer:

76