The Speed Camera

At a glance

Pairs · launched with the derivative’s definition, due the next class · one ticket, one verdict, every rate defended

What you are making

An automated speed camera flashed a car in a $50$ km/h zone, and the ticket in your envelope claims $68$ km/h at the instant of the flash. Your pair also receives what the camera’s maker logged: the car’s position, in metres, every half-second through the zone. The driver’s lawyer has spotted something awkward — an instant has no duration, no distance is travelled at it, and $\frac{0}{0}$ is nobody’s speed. You finish with a verdict — uphold or overturn — built from average speeds over intervals that shrink onto the moment of the flash, and a plain-language letter to the review board explaining how a speed can exist at a single instant at all.

Milestones

• The data plotted by hand; the story of the drive told before a single rate is computed
• Average speeds over at least four shrinking intervals that bracket the flash, approached from both sides
• The march tabulated; the value it settles toward named, with The Limit carrying the argument
• The verdict written for a reader who has never taken calculus, and defended out loud on the due date

How it is assessed

Per How Marks Work, the reasoning is the product: a verdict that reads the shrinking intervals honestly outranks a confident number with no march behind it. On the due date your pair defends the verdict out loud. The Math Journal entry on what sampled data can never quite prove — the car between the samples — completes the evidence.

Success criteria

Quality	What it looks like in your work
Shape before symbols	The drive’s story told from the plot first
An honest march	Intervals shrink onto the flash from both sides
The limit named	One value defended, not an average restated
A humane verdict	The letter convinces a non-mathematician

If the averages will not settle

Check that your intervals genuinely bracket the flash, and shrink them in a pattern — halving works well. If the left and right marches disagree, say so in the verdict: that disagreement is evidence about the data, not a failure of the method.

Curriculum connection

A1.3

make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approximate the slope of the tangent)

Link to original

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 $y=f(x)$?

Link to original

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]

Link to original