>> [music]. Sine is a periodic function.
You follow the graph of sine just wiggles up and down, forever and ever.
Let's try to get a sense as to why this is the case.
Why does the graph of sine look like this? Why is sine moving up and down?
To get some intuition for this, let's first imagine a very different situation.
Let's suppose that velocity is equal to position.
And, I gotta imagine I'm starting here, and I start moving, right?
And I'm going to start moving slowly at first, but as my position increases, I'm
going to be moving faster and faster, right?
As my position is larger and larger, my velocity becomes larger and larger, which
then makes my position larger and larger. So if this is the way the world works, I'm
not bouncing around from minus 1 to 1 like sine.
I'm just being thrown off towards infinity.
We know what that situation looks like. If f of t is my position at time t, then f
prime of t, the derivative of f at t is my velocity at time t.
So saying that velocity equals position is just saying I've got some function whose
derivative is equal to itself. And, we know an example of that.
F of t equals e to the t is one example of such a function.
So, to cook up a different situation, let's imagine something a little bit
different. Instead, suppose that we're in the
situation where my acceleration is negative my position.
So that means that if I'm standing over here, where my position is positive, I'm
to the right of zero, then my acceleration is negative and I'm being pulled back
towards zero. And then, as I swing past toward zero, now
my position is negative, so my acceleration is positive, so I'm being
pulled back in the other direction. And that's kind of producing this swinging
back and forth across 0. This situation comes up physically in the
real world. If you attach a spring to the ceiling, the
spring's acceleration is proportional to negative its displacement.
There's a function also that realizes this model.
To say that acceleration is negative position is as it is the second derivative
of my position, which is my acceleration, is negative my function.
And yeah, we know an example. If f is sine of t, then the derivative of
sin is cosine. And the derivative of cosine, which is the
second derivative of sine, is minus sine of t.
And we know another function just like this.
Look if, if g of t is cosine of t, then g prime of t is minus sine of t.
But then, g double prime of t, the second derivative of cosine, is just minus cosine
of t, because the derivative of sine is cosine.
So cosine is another example of a function whose second derivative is negative
itself. We know even more functions like this.
For instance, what if f were the function f of t equals 17 sine t minus 17 cosine t?
Well then, the derivative of f would be 17 times the derivative of sine, which is
cosine, minus 17 times the derivative of cosine.
But the derivative of cosine is minus sine.
So this is plus 17 sine t. So that's the derivative of f.
The second derivative of f is the derivative of the derivative.
This'll be 17 times the derivative cosine. Which is minus sin, plus the derivative of
sin, which is cosine. So yeah, this is another example of a
function whose second derivative is negative itself.
If I take this function and differentiate it twice, I get negative the original
function. So if you like, the reason why all of
these functions are bouncing up and down like this, is because in every case, the
functions second derivative is negative its value.
When the function is positive, the second derivative is negative, pulling it down.
And when the function's value is negative, the second derivative is positive, pushing
it back up. And so the function bounces up and down
like this forever.