# How can an average rate of change be smaller, yet the function be larger

algebra-precalculuscalculus

Torty, the Tortoise and Harry the Hare question.

Problem Description

[This question] centers around the following situation, describing a 100m race. […]
Situation: Torty and Harry are competing in a 100m sprint race.
Torty's average speed on any 5 second interval is always less than Harry's average speed on any 5 second interval, but Torty wins the race!
(Note: for consistency, let's say they keep running after 100 meters so that their speeds can always be calculated by looking forward in time, but we stop the race at 100 meters.)
Prompt: Discuss as a group (using the definitions of constant and average rate of change) how it is possible that Torty wins the race.
Post an initial conjecture by describing the properties of Torty's and Harry's distance-time relationships that would allow for Torty to win under the constraints of the race.
[…] specifically address how properties of Torty and Harry's distance-time relationships do or do not match with the definitions of constant and average rate of change and and do or do not fit the constraints of the race.
[…] create a neatly drawn or computer generated graph that shows Torty's and Harry's distance-time relationships that visualize the described race.

Some context- I am a tutor. This question has come up at least a couple of times from students. Most recent is a student at Embry Riddle University. Due to constraints with my employer, this is most of what I know about the context, no ability to ask any more clarifying questions, etc. It's probably safe to assume Calculus I methods, so little or no analysis. This is an education context, so the answer should probably incorporate "how their described properties of Torty and Harry's distance time relationships do or do not match with the definition of constant and average rate of change and do or do not fit the constraints of the race."

A more general way (than Ross Millikan's answer) that this can work is something like the following. Torty runs $$11\,\mathrm{ms}^{-1}$$ for one second, then $$1\,\mathrm{ms}^{-1}$$ for four seconds, then repeats. This means averaging $$3\,\mathrm{ms}^{-1}$$ for each $$5$$-second interval, but finishing in just under $$31$$ seconds (covering $$101\,\mathrm{m}$$ in $$31\,\mathrm{s}$$). Harry, meanwhile, runs at a steady speed of $$3.125\,\mathrm{ms}^{-1}$$, beating Torty on any $$5$$-second interval, but finishing in exactly $$32\,\mathrm{s}$$.
After $$30$$ seconds Harry is ahead (inevitably), and after $$35$$ seconds Harry is ahead, but in between Torty is briefly ahead, and this coincides with the winning post.