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>> [music].
Sine is a periodic function.

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You follow the graph of sine just wiggles
up and down, forever and ever.

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Let's try to get a sense as to why this is
the case.

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Why does the graph of sine look like this?
Why is sine moving up and down?

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To get some intuition for this, let's
first imagine a very different situation.

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Let's suppose that velocity is equal to
position.

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And, I gotta imagine I'm starting here,
and I start moving, right?

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And I'm going to start moving slowly at
first, but as my position increases, I'm

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going to be moving faster and faster,
right?

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As my position is larger and larger, my
velocity becomes larger and larger, which

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then makes my position larger and larger.
So if this is the way the world works, I'm

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not bouncing around from minus 1 to 1 like
sine.

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I'm just being thrown off towards
infinity.

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We know what that situation looks like.
If f of t is my position at time t, then f

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prime of t, the derivative of f at t is my
velocity at time t.

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So saying that velocity equals position is
just saying I've got some function whose

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derivative is equal to itself.
And, we know an example of that.

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F of t equals e to the t is one example of
such a function.

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So, to cook up a different situation,
let's imagine something a little bit

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different.
Instead, suppose that we're in the

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situation where my acceleration is
negative my position.

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So that means that if I'm standing over
here, where my position is positive, I'm

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to the right of zero, then my acceleration
is negative and I'm being pulled back

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towards zero.
And then, as I swing past toward zero, now

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my position is negative, so my
acceleration is positive, so I'm being

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pulled back in the other direction.
And that's kind of producing this swinging

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back and forth across 0.
This situation comes up physically in the

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real world.
If you attach a spring to the ceiling, the

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spring's acceleration is proportional to
negative its displacement.

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There's a function also that realizes this
model.

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To say that acceleration is negative
position is as it is the second derivative

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of my position, which is my acceleration,
is negative my function.

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And yeah, we know an example.
If f is sine of t, then the derivative of

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sin is cosine.
And the derivative of cosine, which is the

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second derivative of sine, is minus sine
of t.

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And we know another function just like
this.

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Look if, if g of t is cosine of t, then g
prime of t is minus sine of t.

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But then, g double prime of t, the second
derivative of cosine, is just minus cosine

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of t, because the derivative of sine is
cosine.

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So cosine is another example of a function
whose second derivative is negative

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itself.
We know even more functions like this.

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For instance, what if f were the function
f of t equals 17 sine t minus 17 cosine t?

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Well then, the derivative of f would be 17
times the derivative of sine, which is

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cosine, minus 17 times the derivative of
cosine.

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But the derivative of cosine is minus
sine.

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So this is plus 17 sine t.
So that's the derivative of f.

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The second derivative of f is the
derivative of the derivative.

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This'll be 17 times the derivative cosine.
Which is minus sin, plus the derivative of

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sin, which is cosine.
So yeah, this is another example of a

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function whose second derivative is
negative itself.

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If I take this function and differentiate
it twice, I get negative the original

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function.
So if you like, the reason why all of

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these functions are bouncing up and down
like this, is because in every case, the

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functions second derivative is negative
its value.

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When the function is positive, the second
derivative is negative, pulling it down.

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And when the function's value is negative,
the second derivative is positive, pushing

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it back up.
And so the function bounces up and down

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like this forever.