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

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 $f(a)$

Substitute $x=-4$ into $f(x)$: $f(-4)=4(-4)^{3}-8(-4)^{2}-1=4\times(-64)-8\times16 - 1=-256-128 - 1=-385$.

Step3: Calculate $f(b)$

Substitute $x = 1$ into $f(x)$: $f(1)=4(1)^{3}-8(1)^{2}-1=4 - 8-1=-5$.

Step4: Calculate the average rate of change

$\frac{f(1)-f(-4)}{1-(-4)}=\frac{-5-(-385)}{1 + 4}=\frac{-5 + 385}{5}=\frac{380}{5}=76$.

Answer:

$76$