Imagine a road on which the speed limit is specified at every
single point. In other words, there is a certain function $L$ such that the
speed limit $x$ miles from the beginning of the road is $ L(x)$. Two cars
$A$ and $B$, are driving along this road; car A's position at time $t$ is
$ a(t)$, and car B's is $ b(t)$.
- What equation expresses the fact that the car $A$ always travels at the
speed limit? (Hint: The the answer is {not} $a'(t) = L(t)$.)
- Suppose that $A$ always goes at the speed limit, and that B's position
at time $t$ is A's position at time $t-1$. Show that $B$ is also going at the
speed limit at all times.
- Suppose $B$ always stays at constant distance behind $A$. Under what
conditions will $B$ still always travel at the speed limit?
Tags: diff, chain-rule, abstract-function, proof, math-language, word-problem, t2, c3
Here students must struggle to turn plain English into precise mathematical statements. Because the underlying mathematical equations are not immediately clear, this problem should foster some debate.
– Jeff Barton 1998-08-09
Also see
mike12-001 which extends the problem. –
Eric Hsu 2008-07-12 21:03