Find the average rate of change of \(f( x) =x^{2}+3x\) from \(2\) to \(x;\) \(x\neq 2\).

Find the limit as \(x\) approaches \(2\) of the average rate of change of \(f( x) =x^{2}+3x\) from \(2\) to \(x\).

Solution (a) The average rate of change of \(f\) from \(2\) to \(x\) is \[ \begin{equation*} \dfrac{\Delta y}{\Delta x}=\frac{f(x)-f(2)}{x-2}=\frac{(x^{2}+3x)- [ 2^{2}+3( 2)] }{x-2}=\frac{x^{2}+3x-10}{x-2}=\frac{(x+5)(x-2)}{x-2} \end{equation*} \]

(b) The limit of the average rate of change is \[ \begin{equation*} \lim_{x\rightarrow 2}\frac{f(x)-f(2)}{x-2}=\lim_{x\rightarrow 2}\frac{ (x+5)(x-2)}{x-2}=\lim_{x\rightarrow 2}( x+5) =7 \end{equation*} \]