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[music].
Let's suppose that I have a formula for my

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velocity.
For instance I made my velocity at time t

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is something like 3 minus 10 times t.
With that equation in hand, can I then

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determine my position?
Right?

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Velocity and position are related in some
way.

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The derivative of my position is, well
let's, let's write that fact down.

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P of t is my position at time t.
V of t is my velocity at time t.

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And the crucial thing here is if the
derivative of my position, right?

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The rate of change of my position is
velocity.

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So, the derivative of p is v.
So, I've written that as a derivative

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statement but I could also write it as an
antiderivative statement.

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So, I can write that p of t is an
antiderivative of v of t, right?

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Because p differentiates to v, the
antiderivative of v is p.

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Now, I want to use the fact that my
velocity at time t is 3 minus 10t.

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That means in our specific case that my
position, being the antiderivative of the

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velocity and velocity is 3 minus 10t,
right?

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So, my position is the antiderivative of
my velocity.

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And we can solve that antidifferentiation
problem.

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Well, here we go.
Alright?

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This is the antiderivative of a
difference, which is the difference of

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antiderivatives.
Now, I'd like to antidifferentiate 3,

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would be antiderivative of a constant such
that constant times t.

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And what's an antiderivative of 10 times
t?

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Well, that's 10 times an antiderivative
for t.

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So, I got 3t minus 10 times.
What's an antiderivative for t?

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By the power rule, an antiderivative for t
is t squared over 2.

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And if I differentiate this, I get back t.
I'm going to include a plus C here.

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So I can rewrite this as 3t minus 5t
squared plus some constant C.

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And now, that plus C has a perfectly
reasonable physical interpretation.

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If I know my velocity, I know my position
as long as I know my initial position, as

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long as I know where I started, right?
And that's what that plus C is encoding.

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If I knew for instance that my position at
times 0 were say 4 units, then I could

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figure out what C would have to be right.
That would tell me my position at some

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time t, be 3t minus 5t squared plus 4.
This is an example where knowing my

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initial position, my position at time 0,
and knowing my velocity for all time, is

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enough to figure out my position.
In very fancy terms, this is an example of

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solving a differential equation.
I started with the derivative information,

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and by antidifferentiating, I recovered
the original function, up to a constant.