A Second Train...
Suppose that there is a second train, different from the one we
studied in For example, when t=1 minute, it is
1 kilometer from the station; when t=2 it is 4 kilometers away;
when t=1.1, it is (1.1) |

Here, the information we are given about the motion of this second
train is of a different character. In *Designing a Speedometer*
we had a chart which gives a finite number of readings (each
*reading* is a pair, like t=3 and kilometers=55). Here, we are
essentially given what amounts to *an infinite number of
(odometer) readings*. This is because we can plug in

Now, try to discover what the speedometer of this train reads when t=3. The suggestion is to proceed in several steps in order to make this discovery. Namely:

- First, find the
__average__speed of this train over the time interval "starting when t=3 and ending when t=4" (which we call "the time interval [3,4]").______________________________________________________________________

- Next, find the
__average__speed of this train over the time interval [3, 3.5].______________________________________________________________________

- Then, find it over the interval [3, 3.1].
______________________________________________________________________

- Now, find it over the interval [3, 3+f]
where f is some unknown constant (think of it as a fraction; you just
need to know how to take (3+f)
^{2}and then the rest is the same).______________________________________________________________________ ______________________________________________________________________

Based on (a) through (d), give your (*carefully reasoned out*)
guess as to what the speedometer read at t=3. **Explain your
reasoning** in a paragraph with full sentences.

______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________

It is important to understand that if one asks for a given train
(or any other object), the question: "what is the average speed?" then
the question __only__ makes sense if your question includes a
specification of a time interval, that is, *only* if you ask what
the average speed is *over a time interval which you specify*.
It does not make sense to ask "what is the average speed" but it does
make sense to ask "what was the average speed of the train between
2:00pm and 3:30pm?" (or "between t=3 and t=3.4" if you are using "t"
readings instead of "am" and "pm" actual clock times, which is what we
are doing here).

It is also important to understand that average speed is
not the same as your speedometer's reading:
your speedometer's reading is of course your "instantaneous" or
"instant" speed.
The key is that a collection of average speeds, taken over
smaller and smaller intervals "surrounding" a time t (like t=3) will
give you better and better approximations of the instant speed at
that time t (e.g. at t=3).

**Challenges:** (i) What is this train's speedometer
reading when t=4?
(ii) Suppose a third train has distance
d(t)=t^{2}+1 from the station at time t. Find the speedometer
readings when t=2 and t=3. (iii) A fourth train's
distance from the station at time t is d(t)=3t^{2}+t. What is
the speedometer reading when t=4? Include all your calculations
(neatly organized) and full sentence narration explaining your
reasoning and justifying all of your conclusions!

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