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 (t₂ - t₁).
It's still a distance over a time.
And this distance is how far you traveled during this time.
Okay, 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's 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, "This can't
be right because the 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
and this is not an accurate way of looking at it."
So that's a reasonable objection, actually.
And here's one way we could address it.
You could say, okay, yeah, you're right
the speed might be changing
from 10 to 11, so what if I consider 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 and maybe you stopped for just an instant in
here so that this average speed is not
a good representation, is not a good approximation to this exact speed."
Okay, that's again a reasonable objection.
So all right, fine, I'll try this.
Maybe I'll just calculate the average speed
from t=10 to t=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
(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.