[music]. Let's suppose that I have a formula for my
velocity. For instance I made my velocity at time t
is something like 3 minus 10 times t. With that equation in hand, can I then
determine my position? Right?
Velocity and position are related in some way.
The derivative of my position is, well let's, let's write that fact down.
P of t is my position at time t. V of t is my velocity at time t.
And the crucial thing here is if the derivative of my position, right?
The rate of change of my position is velocity.
So, the derivative of p is v. So, I've written that as a derivative
statement but I could also write it as an antiderivative statement.
So, I can write that p of t is an antiderivative of v of t, right?
Because p differentiates to v, the antiderivative of v is p.
Now, I want to use the fact that my velocity at time t is 3 minus 10t.
That means in our specific case that my position, being the antiderivative of the
velocity and velocity is 3 minus 10t, right?
So, my position is the antiderivative of my velocity.
And we can solve that antidifferentiation problem.
Well, here we go. Alright?
This is the antiderivative of a difference, which is the difference of
antiderivatives. Now, I'd like to antidifferentiate 3,
would be antiderivative of a constant such that constant times t.
And what's an antiderivative of 10 times t?
Well, that's 10 times an antiderivative for t.
So, I got 3t minus 10 times. What's an antiderivative for t?
By the power rule, an antiderivative for t is t squared over 2.
And if I differentiate this, I get back t. I'm going to include a plus C here.
So I can rewrite this as 3t minus 5t squared plus some constant C.
And now, that plus C has a perfectly reasonable physical interpretation.
If I know my velocity, I know my position as long as I know my initial position, as
long as I know where I started, right? And that's what that plus C is encoding.
If I knew for instance that my position at times 0 were say 4 units, then I could
figure out what C would have to be right. That would tell me my position at some
time t, be 3t minus 5t squared plus 4. This is an example where knowing my
initial position, my position at time 0, and knowing my velocity for all time, is
enough to figure out my position. In very fancy terms, this is an example of
solving a differential equation. I started with the derivative information,
and by antidifferentiating, I recovered the original function, up to a constant.