Suppose you are in your car, in the middle of a long drive, while on vacation. Your car has an odometer, and you always make note of its readings when you are at rest stops. At 2pm you are at one rest stop. At 4pm you are at another rest stop.

(a) Suppose you want to know your *average speed* between
2pm and 4pm. What information would use, and, in as precise a way as
you can, explain how exactly you would use it:

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(b) Describe how, in general, you can find your "average speed,
between [starting time] and [ending time]. Use
full sentences, but also be mathematically
precise about *which* quantities you would use, and *how*
you would use them.

Use S for your starting time [the time at which
you would be at the first rest stop] and E for your ending time [the
time at which you arrive at the second rest stop]. Yes, it's ok to
subtract, for example, 4:20pm "minus" 3:30pm" would be "50 minutes"
Your explanation:

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(c) Write down a formula for the average speed between
a starting time (S) and an ending time (E) if your odometer at the
first time equaled D_{1} and at the second time (at the "end")
it equaled D_{2} (since you are driving forward,
D_{2} will be a bigger number than D_{1}).

Formula for average speed, using the above constants and notation:

(d) Now give a more general definition of average speed using
"elapsed time" -- that is, how much time passed between the "first
instant (point in time) we are interested in" and the "second instant"
-- and using "elapsed distance" (this would correspond to the distance
between the two rest stops in our earlier example).

Average Speed = (__________________) ÷ (__________________).

(e) Explain **how** "average speed" is different from "what my
speedometer reads" (the latter gives the *instantaneous speed* at
any *one* point in time when we care to look at it). And,
intuitively, **why** are these two quantities different?

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(f) Explain how "average speed" readings can approximate your car's
instantaneous speed (i.e. its "Speedometer reading" speed). One way to
think about it: Suppose that you were driving your car while a person
in the back seat held a stopwatch in their hands and could tell the
person in the passenger seat to record the odometer at any given time;
whenever the person in the back shouted "Now!" The person in the back
would ask for the odometer reading at exactly 3:00pm and zero
seconds. Then they'd ask for a reading 60 seconds later, and then the
"average speed between 3:00pm and 3:01pm" could be found. Now imagine
the same thing happening, only with a wait of only 30 second between
the two odometer readings. Would the "average speed" computed now be
likely to be closer to the speedometer readings during those 30
seconds? Why? Imagine waiting only 5 seconds between "starting
odometer reading" and ending. Would the "average speed" from
*that* (that is, the car's "average speed" between 3:00pm
exactly, and 5 seconds past 3pm) come even closer to the speedometer
reading? Why?

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