Answer
$a.\quad 17$
$b.\quad-1$
$c.\quad 11$
Work Step by Step
Average rate of change from $a$ to $b$
$=\displaystyle \frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a},\quad a\neq b$
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$a.$
$g(-2)=(-2)^{3}-2(-2)+1=-8+4+1=-3$
$g(-3)=(-3)^{3}-2(-3)+1=-27+6+1=-20$
$\displaystyle \frac{g(-2)-g(-3)}{-2-(-3)}=\frac{(-3)-(-20)}{1}=\frac{17}{1}=17$
$b.$
$g(1)=(1)^{3}-2(1)+1=1-2+1=0$
$g(-1)=(-1)^{3}-2(-1)+1=-1+2+1=2$
$\displaystyle \frac{g(1)-g(-1)}{1-(-1)}=\frac{(0)-(2)}{2}=\frac{-2}{2}=-1$
$c.$
$g(3)=(3)^{3}-2(3)+1=27-6+1=22$
$g(1)=(1)^{3}-2(1)+1=1-2+1=0$
$\displaystyle \frac{g(3)-g(1)}{3-1}=\frac{(22)-(0)}{2}=\frac{22}{2}=11$
