Average-Speed Cameras

Some highways time your car between two cameras miles apart. If your average speed was over the limit, you get a ticket, even if no one saw you speeding. The Mean Value Theorem guarantees that at some instant you were going exactly your average speed.

How it works

The Mean Value Theorem

In one sentence

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then there is a $c \in (a, b)$ with $f'(c) = \frac{f(b) - f(a)}{b - a}$ .

General formula

\frac{{\htmlData{sym=0}{s}}({\htmlData{sym=2}{b}}) - {\htmlData{sym=0}{s}}({\htmlData{sym=1}{a}})}{{\htmlData{sym=2}{b}} - {\htmlData{sym=1}{a}}} = {\htmlData{sym=4}{s'(c)}} \quad \text{for some } {\htmlData{sym=3}{c}} \in ({\htmlData{sym=1}{a}}, {\htmlData{sym=2}{b}})

If your speed was always below your average, you'd have covered less distance than you did. So it must have reached the average somewhere.

What the symbols mean

$s$
Position
Miles traveled since the first camera.
$a$
First camera time
When you pass camera 1.
$b$
Second camera time
When you pass camera 2.
$c$
The guaranteed moment
A time in between when your speed equaled your average.
$s'(c)$
Instantaneous speed
Your speedometer reading at time c.

Your average speed is the slope of the secant line from the first camera to the second.
If position is smooth, some tangent line between them is parallel to that secant.
At that moment $c$ your instantaneous speed $s'(c)$ equals your average speed.
So an average above the limit proves you were above the limit at some instant.

Learn the math

The Mean Value TheoremCalc I · Lesson 32, Applications of Derivatives