Let me generalize this idea of average speed.
So, average speed from time 1 (t₁) to time 2 (t₂),
some time interval along your journey,
is just the distance you traveled
during that time interval from time 1 (t₁) to time 2 (t₂),
divided by how long that time took,
this final time minus the initial time.
It's still a distance over a time.
And this distance is how far you traveled during this time.
OK. So, the problem with this is we still haven't answered the question.
How fast were you going exactly 10 minutes into the ride?
So, the question is how fast were you going exactly 10 minutes into your ride?
not what's your average speed from 9 to 11,
but at exactly t = 10, what is your speed?
Well, we know how to calculate the speed using this formula,
but this formula requires two times, and we're really only given one.
So this is over some time interval from t₁ to t₂,
but I want to know the time at exactly 10.
So, we're in a little bit of a bind,
and calculus, and this idea of a derivative is going to get us out of the bind.
So, here is what we do.
We take a time interval, maybe from t = 10 to t = 11,
and then we calculate the average speed for that.
So, we might say that the speed at exactly t = 10,
is the average speed from t = 10 to t = 11.
But, you might object and say, well, this can't be right
because speed might be changing.
Maybe you're speeding up or slowing down a lot
as you go from minute 10 to minute 11.
So this really isn't your speed at exactly t = 10.
This could be changing. This is not an accurate way of looking at it.
So, that's a reasonable objection actually
and here is one way we could address it.
Say, OK, yeah, you're right, the speed might be changing from 10 to 11,
so what if I insert a smaller interval?
I'll go from 10 to 10.1.
That's just a tenth of a minute, surely the speed isn't changing much.
So the average speed is a good approximation to the speed exactly at t = 10.
And you might object again.
Well, how do you know? Maybe you're speeding up or slowing down a lot.
Maybe you stopped for just an instant in here
so that this average speed is not a good representation,
it's not a good approximation to this exact speed.
OK, well, that's again a reasonable objection.
So, fine, I'll try this.
Maybe I'll just calculate the average speed from t = 10 to 10.01.
And again you might object, well,
yes this is a really small time interval,
but still your speed could be changing in this time interval,
so it's not a good approximation or representation
of the speed at this particular instant.
So, again, I could meet this objection
and we could continue arguing back and forth
and so on and so on and so on,
considering a smaller and smaller and smaller time interval here.
And so, the way out of this bind
-- so it seems we could play this game forever --
is to agree that if the right hand side is getting closer and closer to something,
then we say that's the speed exactly at t = 10.