The average rate of change for a function f(x) from x=a to x=b is:
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
For the given function:
[tex]f(x)=x^3-2x^2+3x[/tex]
Averate rate of change from x=1 to x=3:
[tex]\begin{gathered} \frac{f(3)-f(1)}{3-1}=\frac{(3^3-2(3)^2+3(3))-(1^3-2(1)^2+3(1))}{3-1} \\ \\ =\frac{(27-2(9)+9)-(1-2+3)}{2} \\ \\ =\frac{(27-18+9)-(2)}{2} \\ \\ =\frac{18-2}{2} \\ \\ =\frac{16}{2} \\ \\ =8 \end{gathered}[/tex]Then, the averate rate of change for the given function from x=1 to x=3 is 8
