Welcome to Calculus.
I'm Professor Ghrist. And we're about to begin lecture 13 bonus
material. In our main lesson, we saw that perhaps
the ultimate interpretation of higher derivatives is in terms of Taylor series.
That if we are at a particular input, then the more derivatives we know at that
input, the better we can approximate what the value of the function is farther away
by some step h. It's a bit remarkable that just by
knowing infinitesmal information at a particular input, that is, higher and
higher derivatives at a, you can say what the function is doing far away.
That actually goes fairly deep into other areas of mathematics.
Let's take a few minutes and think of it. And what follows, I going to give you is
the language of operators. Where for purposes of this course, an
operator is something that takes as it's input function and returns as it's output
another function. What are some examples of operators?
Well, there's one that we've been looking at rather closely.
The derivative, I'm going to give it an operator symbol as a capital D.
Then, what does capital D do? It takes a function f and returns the
derivative df dx. Now, what's so special about this?
Well, operators have an algebra associated with them.
For example, we can take D and raise it to certain powers.
We could talk about D squared, as meaning we apply D twice.
You take the derivative, then you take the derivative again.
So, the operator D squared is really taking the second derivative, D cubed is
taking the third derivative. What is D to the 0?
Well, that too has a name. It is called the identity operator, and
is given the special symbol I. The identity operator takes f and returns
f. It does nothing.
That is why it is called the identity. So, d to the zero; the zeroth derivative
of f is of course, f. What happens when we take powers of I?
Well, I squared means do nothing I, and then nothing again.
So, I squared is equal to I. This might remind you of a number with
which you're somewhat familiar, that is the number 1 under multiplication.
1 times 1 is 1. I squared is equal to I.
This identity operator plays the role of the number 1 in the algebra that will
follow. There are a few other interesting
operators that we're going to take a look at.
One, let's call it E, capital E. This corresponds to a shift of f.
E of f is going to be the function which when evaluated at x, gives you f at x
plus 1. This has the effect of shifting the graph
of f to the left by one unit. Now, what happens when we take E squared?
That means shift to the left by one, then shift to the left by one again.
We can talk about E cubed, E to the fourth, etcetera.
We're continually shifting f to the left. Now, can you talk about E raised to a non
integer power? E to the H.
Well, surely that makes sense, we can simply evaluate f not at x, but at x plus
h. And this means of course that we can talk
about E to the negative 1, which we would call a right shift.
Instead of evaluating f at x plus 1, it evaluates at x minus 1.
But note, that this right shift, E inverse really is the inverse of E.
If I multiply E to the negative 1 times E, I get E to the 0 or the identity
operator. And of course, this works whether I
multiply E inverse by E, or E by E inverse.
And this means that I now have a a nice language for doing algebra with these
operators. Let's see what happens when we think in
this language for a little bit. What is E to the h of f evaluated at x.
This is a shift by h. I need to take f and evaluate it at x
plus h. Now, let's use our intuition from Taylor
series. If I take the Taylor expansion of this,
it is the sum k goes from 0 to infinity of 1 over k factorial, times the kth
derivative of f evaluated at x times h to the k.
Now for the moment, let's not worry about where we evaluate this so much.
Let's try to rewrite this in the language of operators.
Now notice, there's a kth derivative of f.
So, let's use our differentiation operator D to the k for taking that kth
derivative of f. And now, we see that there are two terms
that are raised to the kth power. D, the differentiation operator and h,
this perturbation to the input. So, if I collect those two terms together
and say, I've got the sum k goes from 0 to infinity of 1 over k factorial times
quantity hD to the k, I apply that to f and evaluate it at x.
That is what this shift is. Now, step back for a moment and take a
look at what we've done. We've really said that this operator, E
to the h of f, is equal to the sum k goes from 0 to infinity of something to the k
over k factorial applied to f. What is that?
Where have I seen something like that before?
This really looks like an exponential, but what am I exponentiating?
I am exponentiating h times D, the differentiation operator and applying
that to f. If we remove the dependents on f, then
what we see is that this shift operator, capital E, to the h, is precisely the
exponentiation of h times the differentiation operator.
That is really a deep observation. That the shift E, just moving to the left
by one unit is I plus D plus 1 half D squared plus 1 over 3 factorial, D cubed,
et cetera, et cetera. The shift is the exponential of the
differenciation operator. Or, if you like, you can think of the
differentiation operator D as the natural logarithm of the shift operator.
Now, this probably, is causing a little bit of surprise or confusion.
What's going on? Well, don't worry too much.
But think back to how this course began. We asked the question, what is e to the
x? We will continue asking this question
throughout the semester. What we have seen is that there are many
responses to this question, and many layers of understanding what we mean, by
exponentiation. What we mean by differentiation and how
derivatives conspire to give you deep information about a function.