A1.5

make connections, for a function that is smooth over the interval $a\le x\le a+h$, between the average rate of change of the function over this interval and the value of the expression $\frac{f(a+h)-f(a)}{h}$, and between the instantaneous rate of change of the function at $x=a$ and the value of the limit ${\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$ Sample problem: What does the limit ${\lim }_{h\to 0}\frac{f(4+h)-f(4)}{h}=8$ indicate about the graph of the function $f(x)=x^2$? The graph of a general function $y=f(x)$?