U1S1V05 Average Rate of Change.txt

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We're looking for the average velocity of our car
between 8 o'clock and 8:01.
And we were told the positions of the car at those two times.
So f, which is our position function, of 8 is 50 miles.
And f of 8:01, or 8 plus 1/60, is 51 miles.
And from that, hopefully you are able to determine
what the average velocity during this 1 minute was.
So you need delta f, the change in position,
divided by delta t, the change in time.
And if you did that, then on the top
you would get 51 minus 50 miles.
And on the bottom, we have the difference
in time between 8:01 and 8 o'clock.
And so we get 1 mile divided by 1 minute, or 1 mile per minute.
Now, we wanted this in miles per hour.
So there are a couple ways to do that.
One way is to just rewrite everything in terms of hours
rather than minutes.
So on the top we have f of 8 plus 1/60 minus f of 8.
That's the difference in position.
And on the bottom we have the difference in the two
times, so 8 plus 1/60 minus 8.
And that's in hours.
And when you do that, you get 1 mile
on top divided by 1/60 of an hour on the bottom,
and that's 60 miles per hour.
A faster way might have been to just take
this 1 mile per minute and multiply it by a conversion
factor of 60 minutes per hour.
And when we do that, the minutes cancel
and we're just left with 60 miles per hour.
So this gave us our average velocity over this one minute,
or our average rate of change of position with respect to time.
Now, it's important to note that, in this course when
we say average rate of change, it doesn't have
to be with respect to time.
It can be any sort of thing.
For instance, if you have some amount of gas, maybe
some steam, then if you change the temperature of the gas,
then the pressure changes.
So pressure is a function of temperature,
and we could talk about the average rate of change
of pressure with respect to temperature,
as the temperature goes from such and such to such and such.
In general, if you have any function f with some input
variable x, then you can talk about the average rate
of change of f of x with respect to x, as x goes
from x equals a to x equals b.
And what this is, is exactly what we had above.
It's just the change in f, so delta
f, divided by the change in x, delta x.
And this is always going to be measured
in units of the output divided by units of the input.
And the formula is just as follows.
So delta x-- x is going from a to b,
so delta x, the change in x, is going to be b minus a.
And delta f, well, how much does f change?
f is going from f of a to f of b, so its change is
f of b minus f of a.
And so this quotient right here is our average rate
of change of f with respect to x as x goes from x equals a to x
equals b.