Introducing Calculus: Can Change Occur at an Instant?

A car's speedometer shows one number at one moment: 50 km/h right now. But speed is distance divided by time, and in a single instant the car covers no distance in no time. Around 450 BCE the Greek philosopher Zeno of Elea pushed this into a paradox: at any one instant a flying arrow occupies a space exactly its own size, so at that instant it is not moving, and yet it gets to the target.

Average rate of change

When a quantity $f(x)$ changes as $x$ goes from $a$ to $b$, its average rate of change on $[a, b]$ is the change in output divided by the change in input:

Its units are output units per input unit: meters per second, gallons per minute, dollars per year. On a graph it is the slope of the secant line, the straight line through the two points $(a, f(a))$ and $(b, f(b))$.

Take a ball tossed straight up from the ground. Rounding gravity to 10 m/s², its height after $t$ seconds is $s(t) = 20t - 5t^2$ meters, so $s(1) = 15$, $s(2) = 20$ and $s(3) = 15$. Its average velocity on $[1, 2]$ is

On $[1, 3]$ the average velocity is $\frac{15 - 15}{2} = 0$ m/s, even though the ball was never sitting still: it rose for one second and fell back for one second. An average rate describes the interval as a whole and erases what happened inside it. The longer the interval, the more it erases.

Why an instant breaks the formula

To get the velocity at exactly $t = 1$, the obvious move is to make both endpoints equal to 1:

This is not 0 and it is not a very large number. It is undefined, because division by zero has no answer. And $\frac{0}{0}$ is undefined in a particularly unhelpful way: the equation $0 \cdot q = 0$ holds for every number $q$, so no single number deserves to be called the quotient. The average-rate formula needs the input to change. At a single instant it doesn't, so the formula has nothing to say there, and this is Zeno's arrow in algebra.

Instantaneous rate as a limit of average rates

The way out is to use intervals that contain the instant and shrink toward it. Each one has a perfectly good average rate, and you watch where those averages head. For the ball at $t = 1$, use the interval from 1 to $1 + h$, where $h$ is a small number, positive (the interval extends to the right of 1) or negative (to the left).

From both sides the averages close in on 10. Algebra shows why. For any $h \neq 0$,

Cancelling $h$ is legal precisely because $h \neq 0$; the average is never computed over an interval of length zero. But the expression $10 - 5h$ makes it plain that when $h$ is close to 0, the average velocity is close to 10, and it can be brought as close to 10 as you like by taking $h$ small enough. That target value is the instantaneous velocity: 10 m/s upward at $t = 1$. The averages from the right are below 10 and those from the left are above 10 because the ball is slowing down, so a later interval catches it moving more slowly.

On the graph, each average rate is the slope of a secant line through $(1, 15)$. As the second point slides toward $(1, 15)$, the secant lines turn and settle on one line, the tangent line, whose slope is the instantaneous rate.

The notation $\lim\limits_{h \to 0}$ reads "the limit as $h$ approaches 0" and asks where the values head as $h$ gets close to 0, never what happens at $h = 0$ itself. Topic 1.2 makes the notation precise, and in Unit 2 this same limit becomes the derivative.

When there is no single rate

The averages don't always agree. For $f(x) = |x|$ at $x = 0$, the average rate on $[0, h]$ is $\frac{|h| - 0}{h}$. For every $h > 0$ this equals $1$, and for every $h < 0$ it equals $-1$. Intervals to the right all say the rate is 1, intervals to the left all say it is $-1$, and shrinking them never brings the two answers together. So $|x|$ has no instantaneous rate of change at 0; its graph has a sharp corner there, and no single tangent line. The instantaneous rate exists only when the average rates on both sides approach the same number, which is why the definition checks both.

Worked examples

Where students lose points