Welcome to Calculus.
I'm Professor Greist and we're about to begin Lecture 16, Bonus material.
Let's turn challenge problem, and consider the Infinite Power Tower.
What is the Infinite Power Tower? I'm so glad you asked.
It is x to the x, to the x, to the x, to the x, to the x, to the x, to the x, on
and on, add infin item. Now, your reaction is likely one of
disbelief. What is such a function, what does that
even mean? How can that exist?
That, that can't be a real function. Well, not only is it a real function, but
we are going to compute its derivative. The way we're going to do so is
implicitly the first step is to give it a name.
Let's call it y. How is a name going to help us compute
the derivative of this function? Well, let's think for a moment.
Instead of giving you an explicit formula for y as a function of x, let's rewrite
it as follows. y equals x to the y.
This is an implicit definition, where I'm defining why in terms of its relationship
to x and to itself. And now, with this as the alternate
definition, let's compute the derivative, dy dx.
We can use the methods of this lesson to do so.
Take the logarithm of both sides of the equation, what do we get?
Well, we get log of y equals y times log of x.
And now, differentiating both sides, gives us what on the left dy over y.
On the right, using the product rule, we get y over x dx plus log of x dy.
And now, the rest is just a little bit of Algebra.
Move terms over to the left, collect, factor out the dy.
What's left over on the right is y over x dx.
And now, rearranging term, we solve for dy dx as y over x times 1 over y minus
log of x to the negative 1. If you like, you can rearrange the terms
a little bit, get y squared over x times quantity 1 minus y log x.
Now, notice there's no way we're going to be able to solve this as a function of x
itself. We, we can't eliminate those y's
completely. We could substitute back in the Infinite
Power Tower. And that would make our formula a little
bit messy looking. But consider how easy it was to do this
computation and get a formula for the derivative.
Now, we're still not done. We haven't answered the question, does
this function actually exist? If, if it does, does its derivative exist
just because you can compute something that seems to be a derivative, does not
mean that the function is, in fact, differentiable.
There are limits to be computed in order to prove this.
Well, we're not going to do all of those details, but if you're curious, it is, in
fact, true that this function exists. It is well defined and differntiable on a
certain domain. That domain being x between e to the
negative e and e to the e minus 1. The function itself, it looks well, a
little complicated. It, it blows up pretty quickly after the
value of x equals 1, where the function, of course, takes on a value of 1.
And 1 to the 1 to the 1 to the 1 to the 1, etcetera.
But what is perhaps a, a bit more surprising is that there is some interval
around 1, where this function can be analyzed where its derivative can be
computed. If you talk about its Taylor series,
etcetera. In fact, this Infinite Power Tower is
related to some very interesting and useful functions.
Functions you won't normally see in an undergraduate class.
The Lambert W function is perhaps the most useful function associated to this
Infinite Power Tower. It comes up in certain models of biology
and population dynamics. We won't be seeing any of those details
or really anything more from this function during the rest of this course.
But I want you to be introduced to the infinite power tower.
Why? To give you nightmares, well, no.
What I want you to see is that with your mind alone, and the tools that you're
learning, you can analyze functions that you cannot see or feel.
You cannot type the infinite power tower into your graphing calculator.
You don't know anything about it except its implicit definition.
Yet, you are still able to compute its derivative.
Quickly and simply. That is a remarkable thing.
That is what Mathematics is all about.