Answer
a)The average rate of change of $f\left( x \right)=-2{{x}^{2}}+4$ from $0$ to $2$ is $-4$.
b)The average rate of change of $f\left( x \right)=-2{{x}^{2}}+4$ from $1$ to $3$ is $-8$.
c)The average rate of change of $f\left( x \right)=-2{{x}^{2}}+4$ from $1$ to $4$ is $-10$.
Work Step by Step
(a)
The average rate of change of $f\left( x \right)=-2{{x}^{2}}+4$ from $0$ to $2$ is
$=\,\frac{\Delta y}{\Delta x}=\frac{f\left( 2 \right)-f\left( 0 \right)}{2-0}\,\,$.
$f\left( 2 \right)=-2{{\left( 2 \right)}^{2}}+4=-8+4=-4$
$f\left( 0 \right)=-2{{\left( 0 \right)}^{2}}+4=0+4=4$.
Average rate of change $=\frac{-4-4}{2-0}$,
$=\frac{-8}{2}\,$,
$=-4$.
(b)
The average rate of change of $f\left( x \right)=-2{{x}^{2}}+4$ from $1$ to $3$ is
Average rate of change $=\,\frac{\Delta y}{\Delta x}=\frac{f\left( 3 \right)-f\left( 1 \right)}{3-1}\,\,$.
$f\left( 3 \right)=-2{{\left( 3 \right)}^{2}}+4=-18+4=-14$
$f\left( 1 \right)=-2{{\left( 1 \right)}^{2}}+4=-2+4=2$.
Therefore, the average rate of change $=\frac{-14-2}{3-1}$,
$=\frac{-16}{2}$,
$=-8$.
(c)
The average rate of change of $f\left( x \right)=-2{{x}^{2}}+4$ from $1$ to $4$ is
Average rate of change $=\,\frac{\Delta y}{\Delta x}=\frac{f\left( 4 \right)-f\left( 1 \right)}{4-1}\,$.
$f\left( 4 \right)=-2{{\left( 4 \right)}^{2}}+4=-32+4=-28$ $f\left( 1 \right)=-2{{\left( 1 \right)}^{2}}+4=-2+4=2$
Thus, average rate of change $=\frac{-28-2}{4-1}$
$=\frac{-30}{3}$
$=-10$
