Example

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 $\begin{equation*} \lim_{x\rightarrow 2}\frac{f(x)-f(2)}{x-2}=\lim_{x\rightarrow 2}\frac{ (x+5)(x-2)}{x-2}=\lim_{x\rightarrow 2}( x+5) =7 \end{equation*}$