Question

find the average rate of change of $f(x)=9x^{3}-20x^{2}-8x$ over the interval $-1,3$. write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$, $b = 3$, and $f(x)=9x^{3}-20x^{2}-8x$.

Step2: Calculate $f(3)$

$f(3)=9\times3^{3}-20\times3^{2}-8\times3=9\times27-20\times9 - 24=243-180 - 24=39$.

Step3: Calculate $f(-1)$

$f(-1)=9\times(-1)^{3}-20\times(-1)^{2}-8\times(-1)=-9 - 20 + 8=-21$.

Step4: Calculate the average rate of change

$\frac{f(3)-f(-1)}{3-(-1)}=\frac{39-(-21)}{3 + 1}=\frac{39 + 21}{4}=\frac{60}{4}=15$.

Answer:

15