The Getaway Car
Reconstructing a Chase from Rate Data

A rectangle uses only one height per slice: the left reading, or the right reading. What if, for each slice, you used the average of its two end readings as the height instead? Recompute the total distance for all six slices this way.

Sketch this new shape sitting under the curve for one slice. What shape is it, if not a rectangle?

Compare this new total to your two Phase 1 totals. Where does it sit, exactly, relative to the left-based and right-based totals?

Write your left sum and your right sum in general terms, using \(v_0, v_1, \ldots, v_6\) for the seven readings and \(\Delta t = 10\). Add the two sums together and divide by two. Show algebraically that this always equals the slice-by-slice averaging method from Task 2a, for any set of readings, not just this one.

What you've just derived is called the Trapezoidal Rule. For \(n\) equal slices with readings \(v_0, v_1, v_2, \ldots, v_n\) and slice width \(\Delta t\), write the general formula for the total estimate. Check that it matches the pattern you found in Task 2c: does every interior reading get counted once or twice?

Every method you've used so far still pretends the speed does something simple, constant, or a straight-line average, across a 10-second gap. Imagine shrinking every slice's width toward zero while using infinitely many of them. What happens to the accuracy of your estimate? Would you still expect a gap between a left-based version and a right-based version?

Mathematicians write this idea with a stretched capital S, which literally means "sum," left over from how Leibniz first drew it. If \(dt\) stands for an infinitesimally thin slice of time and \(v(t)\) is the height of the strip at time \(t\), here is the notation for the exact total distance:

Investigators recover a second component from the wreck: the engine control module, which logs speed continuously rather than every 10 seconds. Enter your seven \((t, v)\) pairs into your GDC and run a quadratic regression. Record the equation it gives you for \(v(t)\), and record the \(R^2\) value it reports.

Use your GDC's numerical integration feature to evaluate \(\displaystyle\int_0^{60} v(t)\,dt\) with your regression equation in place of \(v(t)\). Record the value. This is the closest thing to an exact total distance available to your team.

Line up all four of your distance estimates: left rectangles, right rectangles, trapezoids, and the GDC's integral of your regression model. Police say the getaway car needed to travel at least 1100 m in this 60-second window to reach the northbound on-ramp.

Which of your estimates prove that on their own? Which don't? What does this disagreement tell you about relying on a single rough estimate when the stakes are this high?

Look at the \(R^2\) value your GDC reported for the regression. Where does your quadratic model disagree most with the original black box readings? Give a real-world reason a car's actual speed might not follow a perfectly smooth curve.

Suppose the driver had slammed the brakes and accelerated repeatedly within a single 10-second window, instead of speeding up and slowing down smoothly. Would rectangles or trapezoids still give you an estimate you could trust in court? What would you need instead?

Without using the word "area," explain in your own words what \(\displaystyle\int_0^{60} v(t)\,dt\) is actually computing. What is it adding up, and why does the width of every piece need to shrink all the way to zero for the answer to be exact rather than approximate?