what is the average rate of change of the function f(x)=-x²/4 + 7 over the interval -6 ≤ x ≤ 0? 3/2 1 1/2 5/2

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, $a=-6$, $b = 0$, and $f(x)=-\frac{x^{2}}{4}+7$.

Step2: Calculate $f(a)$

Substitute $x=-6$ into $f(x)$: $f(-6)=-\frac{(-6)^{2}}{4}+7=-\frac{36}{4}+7=-9 + 7=-2$.

Step3: Calculate $f(b)$

Substitute $x = 0$ into $f(x)$: $f(0)=-\frac{0^{2}}{4}+7=7$.

Step4: Calculate average rate of change

$\frac{f(0)-f(-6)}{0-(-6)}=\frac{7-(-2)}{6}=\frac{7 + 2}{6}=\frac{9}{6}=\frac{3}{2}$.

Answer:

$\frac{3}{2}$