Average and instantaneous rates of change

Average rate of change
(ARoC)		 Instantaneous rate of change
(IRoC)
\(\begin{aligned} \text{ARoC} &= \dfrac{\Delta y}{\Delta x} \\ &= \dfrac{f(b)-f(a)}{b-a} \\ &= \dfrac{f(a+h)-f(a)}{h} \end{aligned}\) \(\begin{aligned} \text{IRoC} &= \lim\limits_{\Delta x\to 0}\dfrac{\Delta y}{\Delta x} \\ &= \lim\limits_{b\to a}\dfrac{f(b)-f(a)}{b-a} \\ &= \lim\limits_{h\to 0}\dfrac{f(a+h)-f(a)}{h} \end{aligned}\)
Equivalent to the slope of the secant line between the two points \((a,f(a))\) and \((b,f(b))\) Equivalent to the slope of the tangent line at point \((a,f(a))\)
Depends on those two points, not on the shape of the graph between them. This means that the ARoC between \((a,f(a))\) and \((b,f(b))\) is the same for any function passing through these two points. Depends on the shape of the graph around that one point. This means that the IRoC at \((a,f(a))\) depends on the exact function that passes through that point.

The IRoC at \(x=a\) can be approximated as the ARoC over a small interval, preferably centred, around \(x=a\). We typically calculate the ARoC between \(x=a-0.01\) and \(x+0.01\) or between \(x=a-0.001\) and \(x=a+0.001\).

Difference quotient

The difference quotient of a function \(f\) is the average rate of change defined between arbitrary values \(a\) and \(b\) of its argument, where \(h=b-a\)\[\dfrac{f(b)-f(a)}{b-a}=\dfrac{f(a+h)-f(a)}{h}\]

The black dot represents the \(a\) value in the difference quotient. The green dot is \(h\) to the left, and the teal dot is \(h\) to the right of \(a\). The green dashed line is the secant through \((a,f(a))\) and \((a-h,f(a-h))\), the teal dashed line is the secant through \((a,f(a))\) and \((a+h,f(a+h))\), and the read dashed line is the secant through \((a-h,f(a-h))\) and \((a+h,f(a+h))\). Therefore, the slope of the green line is the AROC on an interval to the left, that of the teal line is the AROC on an interval to the right, and that of the red line is the AROC on a centred interval.
Try changing the values of \(a\) and \(h\) by moving the sliders, and observe the secant lines (and their slopes). They diverge or converge depending on both the location of \(a\) and the size of \(h\). If you choose large enough \(h\) that takes you outside the domain, the interval (on the \(x\)-axis) will be red, and the AROC is impossible to calculate.
At any point, you may press the “Reset” button to return to \(a=2\) and \(h=0.1\).