find the average rate of change for the function over the given interval. y = 5x³ + 8x² - 4 between x = - 9…

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, $a=-9$, $b = 3$, and $f(x)=5x^{3}+8x^{2}-4$.

Step2: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$: $f(3)=5\times3^{3}+8\times3^{2}-4=5\times27 + 8\times9-4=135+72 - 4=203$.

Step3: Calculate $f(-9)$

Substitute $x=-9$ into $f(x)$: $f(-9)=5\times(-9)^{3}+8\times(-9)^{2}-4=5\times(-729)+8\times81 - 4=-3645+648-4=-2996 - 4=-3001$.

Step4: Calculate the average rate of change

$\frac{f(3)-f(-9)}{3-(-9)}=\frac{203-(-3001)}{3 + 9}=\frac{203 + 3001}{12}=\frac{3204}{12}=267$.

Answer:

D. 267