Answer
a)The average rate of change of $f\left( x \right)=-{{x}^{3}}+1$ from $0$ to $2$ is $-4$.
b)The average rate of change of $f\left( x \right)=-{{x}^{3}}+1$ from $1$ to $3$ is $-13$.
c)The average rate of change of $f\left( x \right)=-{{x}^{3}}+1$ from $-1$ to $1$ is $-1$.
Work Step by Step
(a)
The average rate of change of $f\left( x \right)=-{{x}^{3}}+1$ from $0$ to $2$ is
Average rate of change $=\,\frac{\Delta y}{\Delta x}=\frac{f\left( 2 \right)-f\left( 0 \right)}{2-0}\,\,$.
$f\left( 2 \right)=-{{\left( 2 \right)}^{3}}+1=-8+1=-7$
$f\left( 0 \right)=-{{\left( 0 \right)}^{3}}+1=0+1=1$.
Thus, average rate of change $=\frac{-7-1}{2-0}$
$=\frac{-8}{2}\,$
$=-4$
(b)
The average rate of change of $f\left( x \right)=-{{x}^{3}}+1$ from $1$ to $3$ is
Average rate of change $=\,\frac{\Delta y}{\Delta x}=\frac{f\left( 3 \right)-f\left( 1 \right)}{3-1}\,\,$.
$f\left( 3 \right)=-{{\left( 3 \right)}^{3}}+1=-27+1=-26$
$f\left( 1 \right)=-{{\left( 1 \right)}^{3}}+1=-1+1=0$.
Therefore, average rate of change $\,=\frac{-26-0}{3-1}\,\,\,$
$=\frac{-26}{2}\,$
$=-13$.
(c)
The average rate of change of $f\left( x \right)=-{{x}^{3}}+1$ from $-1$ to $1$ is
Average rate of change $=\,\frac{\Delta y}{\Delta x}=\frac{f\left( 1 \right)-f\left( -1 \right)}{1-\left( -1 \right)}\,$.
$f\left( 1 \right)=-{{\left( 1 \right)}^{3}}+1=-1+1=0$
$f\left( -1 \right)=-{{\left( -1 \right)}^{3}}+1=-\left( -1 \right)+1=2$.
Thus, average rate of change $=\frac{0-2}{1-\left( -1 \right)}$,
$=-\frac{2}{2}$,
$=-1$.
