Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The average velocity of an object is its change in position divided by the total amount of time taken. When calculating the average velocity, only the times and positions at the starting and ending points are taken into account. The average velocity is different from the instantaneous velocity, and the two are many times not the same number.
I want to talk about average velocity.
Let's start by doing an example.
A pumpkin is catapulted into the air, time
T as in seconds and the height of
the pumpkin F of T is feet and it's
given by values on this table.
I've got values for T equals 0 to 8.
Now I want to calculate some changes
in position.
And get used to this symbol in calculus,
the delta means change in.
So delta position means change in position.
Over the interval from 0 to 1. So the change
of position from on the interval
of 0 to 1 would be F of 1 minus
F of 0. Now, F of 1 is 118.
F of 0 is 4. And so I'd get 114.
But remember the units are feet.
So this would be in feet.
So that means that the pumpkin over that
first second increases in height 114 feet.
Now let's look at the interval from 1
to 4. The change in position F of
4 minus F of 1. Also remember it's
final position minus initial
position when you're calculating
a change in position.
F of 4 is 268.
And again F of 1 is 118.
So you just subtract.
And you get 150 feet.
And finally from 4 to 6, you get F
of 6 minus F of 4. F of 6, 208.
That's the final position.
Minus the initial position.
F of 4, 268.
And you get negative 60.
And what does that mean?
That means that over this
interval from T.
Equals 4 to T equals 6 the pumpkin
has dropped 60 feet.
What does this have to do with average velocity?
Average velocity over an interval from A to
B is defined as the change in position
divided by the change in time.
And so what we were calculating just
now were changes in position.
If you divide that by the change in time,
the length of the interval, you get
the average velocity.Let's compute some average velocities.
First, for the interval of 1 to 4 we would
have average velocity is F of 4 minus
F of 1 over 4 minus 1. Now we
already calculated F of 4 minus
F of 1. It's back here. 150 feet.
So this would be on top 150 feet
and on bottom 3 seconds.
So this is going to be 50 feet per second.
Okay.
And how about on the interval of 5 to 7?
Average velocity is F of 7. Remember,
final position minus initial.
Minus F of 5 over 7 minus 5. F of
7 going back to the table is 130.
F of 5, 254.
So we have 130 minus 254 over 7 minus
-- actually, I'll just write 2. 7
minus 5 is 2. Now, this numerator,
130 minus 254 is negative 124.
And that's in feet. Over two seconds.
And that's going to give me
minus 62 feet per second.
What does it mean that the average
velocity is negative?
It just means that the pumpkin's probably
on its way down on this interval of
time.
So average velocity is change in
position over change in time.
And remember change in position is final
position minus initial position.
Unit
The Derivative