A1.6

compare, through investigation, the calculation of instantaneous rates of change at a point $(a,f(a))$ for polynomial functions [e.g., $f(x)=x^2$, $f(x)=x^3$], with and without simplifying the expression $\frac{f(a+h)-f(a)}{h}$ before substituting values of $h$ that approach zero [e.g., for $f(x)=x^2$ at $x=3$, by determining $\frac{f(3+1)-f(3)}{1}=7$, $\frac{f(3+0.1)-f(3)}{0.1}=6.1$, $\frac{f(3+0.01)-f(3)}{0.01}=6.01$, and $\frac{f(3+0.001)-f(3)}{0.001}=6.001$, and by first simplifying $\frac{f(3+h)-f(3)}{h}$ as $\frac{(3+h{)}^2-3^2}{h}=6+h$ and then substituting the same values of $h$ to give the same results]