U1S1V04 Average Velocity.txt

#
# File:   content-mit-18-01-1x-captions/U1S1V04 Average Velocity.txt
#
# Captions for MITx 18.01.1x module [BBvacxW-h7M]
#
# This file has 62 caption lines.
#
# Do not add or delete any lines.  If there is text missing at the end, please add it to the last line.
#
#----------------------------------------

OK, we've calculated the average velocity
of our car between 8:00 and 10:00.
But then we decided that that might not
be such a good approximation if we wanted
the velocity at 8:00 exactly.
So we've decided that instead, we
should do the same calculation just this time,
between 8:00 and 8:01.
All right, we're going to have a lot of numbers floating around
here.
So let's get some notation to organize all this.
We know that position is a function of time.
So if we have t as representing time,
then we can say that f of t is position.
So our initial data was that our position at 8:00 was 50 miles.
So f of 8 equals 50.
And our position at 10:00 was 220 miles.
Now previously, you had used this data
to calculate that the average velocity between 8:00 and 10:00
was 85 miles per hour.
And that 85 came from 220 minus 50.
So we traveled 170 miles.
And then you divide by 10 minus 8.
That was the time, two hours, that the journey took.
So in our new notation, this is f of 10 minus f of 8
on top divided by 10 minus 8.
So our numerator here is the change in position.
And our denominator is the change in time.
And when you divide those two, we
get our average velocity of 85 miles per hour.
Now, calculus is all about variables changing, just
all over the place.
And so we have a special notation
that we use to denote the change in a variable.
So here, this numerator where we're
saying the change in position, or the change in f,
we often denote that by Delta f.
So Delta, this Greek letter here, this triangle,
stands for difference.
And so we have the difference in f.
This is not Delta times f.
Delta isn't a thing in and of itself.
It's just one quantity, Delta f.
And similarly, on the denominator,
we're going to have a quantity, Delta t, the change in time.
So this Delta f divided by Delta t
is giving us our average velocity over the period
of time from 8:00 to 10:00.

Now, we wanted to talk about 8:01.
So in order to do that, I need to tell you the position
of the car at 8:01.
And in our notation, that's going to be f of 8
plus 1 over 60.
So that's 8 and 1/60 hours.
And let's say that the car was at mile marker 51
at that moment.
So given that information, what would you say
is the average velocity of the car between 8:00 and 8:01?
Why don't you think about that, and then
we'll come back and discuss some more.