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

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

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

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

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

Â Let's suppose that velocity is equal to position.

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

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

Â going to be moving faster and faster, right?

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

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

Â not bouncing around from minus 1 to 1 like sine.

Â I'm just being thrown off towards infinity.

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

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

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

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

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

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

Â different. Instead, suppose that we're in the

Â situation where my acceleration is negative my position.

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

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

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

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

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

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

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

Â spring's acceleration is proportional to negative its displacement.

Â There's a function also that realizes this model.

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

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

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

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

Â second derivative of sine, is minus sine of t.

Â And we know another function just like this.

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

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

Â of t, because the derivative of sine is cosine.

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

Â itself. We know even more functions like this.

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

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

Â cosine, minus 17 times the derivative of cosine.

Â But the derivative of cosine is minus sine.

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

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

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

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

Â function whose second derivative is negative itself.

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

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

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

Â functions second derivative is negative its value.

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

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

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

Â like this forever.

Â