what is the average rate of change for the function (f(x)=-2x^{2}+5) over the interval (-3.5leq xleq0)?\na…

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, $a=-3.5$, $b = 0$, and $f(x)=-2x^{2}+5$.

Step2: Calculate $f(a)$ and $f(b)$

First, find $f(-3.5)$: [ \begin{align*} f(-3.5)&=-2\times(-3.5)^{2}+5\ &=-2\times12.25 + 5\ &=-24.5+5\ &=-19.5 \end{align*} ] Then, find $f(0)$: [ f(0)=-2\times0^{2}+5 = 5 ]

Step3: Calculate the average rate of change

[ \begin{align*} \frac{f(0)-f(-3.5)}{0-(-3.5)}&=\frac{5-(-19.5)}{3.5}\ &=\frac{5 + 19.5}{3.5}\ &=\frac{24.5}{3.5}\ &=7 \end{align*} ]

Answer:

B. 7