Solution:
The average rate of change is given by;
[tex]\frac{\Delta h}{\Delta x}=\frac{h_2-h_1}{x_2-x_1}_{}[/tex]Given:
[tex]h(x)=3x^3-3x^2-2[/tex][tex]\begin{gathered} h_2=h(1) \\ h_1=h(-2) \\ x_2=1 \\ x_1=-2 \end{gathered}[/tex][tex]\begin{gathered} h(1)\text{ means susbstituting when x =1 into h(x);} \\ h(x)=3x^3-3x^2-2 \\ h(1)=3(1^3)-3(1^2)-2 \\ h(1)=3(1)-3(1)-2 \\ h(1)=3-3-2 \\ h(1)=-2 \end{gathered}[/tex][tex]\begin{gathered} h(-2)\text{ means susbstituting when x =-2 into h(x);} \\ h(x)=3x^3-3x^2-2 \\ h(-2)=3(-2^3)-3(-2^2)-2 \\ h(-2)=3(-8)-3(4)-2 \\ h(-2)=-24-12-2 \\ h(-2)=-38 \end{gathered}[/tex][tex]\begin{gathered} \frac{\Delta h}{\Delta x}=\frac{h_2-h_1}{x_2-x_1}_{} \\ \frac{\Delta h}{\Delta x}=\frac{h_2-h_1}{x_2-x_1}_{}=\frac{h(1)-h(-2)}{x_2-x_1} \\ \frac{\Delta h}{\Delta x}=\frac{-2-(-38)_{}}{1-(-2)} \\ \frac{\Delta h}{\Delta x}=\frac{-2+38}{1+2} \\ \frac{\Delta h}{\Delta x}=\frac{36}{3} \\ \frac{\Delta h}{\Delta x}=12 \end{gathered}[/tex]Therefore, the average rate of change of h(x) from x = -2 to x = 1 is 12.