U2S4V03 Changing Variables, Changing Units.txt

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So we've been thinking about these variables--
x measured in meters, u and t being time variables measured
in seconds and minutes respectively.
And we had that x was a function of u.
Specifically, it was 2u plus 100.
And so we have that the derivative is just 2.
Now, that's meters per second.
What if we really want t as our time variable?
Well, we discovered that u can just be written as 60 times t,
and the 60 represents 60 seconds per minute.
So it's a conversion factor just between minutes and seconds,
and you can call that du dt if you want.
Now, if you plug in 60t for u, then we
get x written as a function of t.
So it's 120t plus 100.
And so this derivative is 120, and that's
in meters per minute.

Why are we doing this?
So it seems pretty simple.
What I want to point out is the following.
So we have x equaling g of t, but it also equals f of u.
OK.
So far, so good.
But note g prime of t, we said that that was 120,
and f prime of u was 2.
So those aren't the same number.
What's going on?
Well, it's not so hard to explain using Leibniz notation.
So g prime is dx dt, whereas f prime is dx du.
And we know that dx dt and dx du aren't
measuring the same things.
So we've switched our input variable from t to u,
and when you do that your derivatives have
to adjust as well.
Now, we've mentioned this before.
This is one of the nice things about Leibniz notation
is that it alerts us to things like this.
But what we want to do today is identify exactly how we need
to adjust our derivatives when we switch our input
variable from t to u.
So dx dt, we said, was 120 meters per minute.
dx du was 2 meters per second.
And the relationship between them
is exactly our conversion factor.
We can just multiply 2 meters per second
by 60 seconds per minute.
So we can write dx dt as equal to dx du times the conversion
factor, which is du dt.
We can think of x depending on t and get this derivative dx dt,
or we can think of x as depending on u
and get this derivative dx du.
And they differ by exactly this conversion factor,
which is how fast u changes with respect to t.
Now, all of these here were linear functions.
x was a linear function of u.
u was a linear function of t.
You might ask if this still works when our functions are
not linear, when they're curvy.
But one great thing about derivatives
is that they allow us to do linear approximation.
That is, near a point, we can approximate a curvy function
using a linear function.
And so this basic principle of conversion
is actually going to hold much more generally,
and we have some exercises to help
you explore just how that works.