Finding the Average Rate of Change

Find the average rate of change of \(f( x) =3x^{2}\):

1. From \(1\) to \(3\)
2. From \(1\) to \(x\), \(x\neq 1\)

Solution (a) The average rate of change of \(f( x) =3x^{2}\) from \(1\) to \(3\) is \[ \dfrac{\Delta y}{\Delta x}=\dfrac{f( 3) -f( 1) }{3-1}= \dfrac{27-3}{3-1}=\dfrac{24}{2}=12 \]

See Figure 17.

Notice that the average rate of change of \(f( x) =3x^{2}\) from \( 1 \) to \(3\) is the slope of the line containing the points \(( 1,3) \) and \(( 3,27) .\)

(b) The average rate of change of \(f( x) =3x^{2}\) from \(1\) to \(x\) is \[ \begin{eqnarray*} \dfrac{\Delta y}{\Delta x}&=&\dfrac{f( x) -f( 1) }{x-1}= \dfrac{3x^{2}-3}{x-1}=\dfrac{3( x^{2}-1) }{x-1}\nonumber \\ &=&\dfrac{3(x-1) ( x+1) }{x-1}=3( x+1) =3x+3 \end{eqnarray*} \]

provided \(x\neq 1\).