## Calculus (3rd Edition)

Average rate of change=$\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$ Instantaneous rate of change=$\lim\limits_{\Delta x \to 0}$(average rate of change) $=\lim\limits_{\Delta x \to 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}=\lim\limits_{\Delta x \to 0}\frac{f(2+\Delta x)-f(2)}{\Delta x}$ $=\lim\limits_{\Delta x \to 0}\frac{[3(2+\Delta x)^{2}-5]-[3(2)^{2}-5]}{\Delta x}=\lim\limits_{\Delta x \to 0}\frac{12+3(\Delta x)^{2}+12\Delta x-5-12+5}{\Delta x}$ $=\lim\limits_{\Delta x \to 0}3\Delta x+12=12$