Welcome to Calculus. I'm Professor Ghrist. We're about to begin Lecture 57,
Bonus Material. Well that seems like a bad way to end the
course, talking about what we can't do. Let's change things up a little bit. Go back to the very
beginning of this class that began with the question
what is e to the x? Well by now you all know
exactly what e to the x is and means on so many different levels,
but lets reconsider this question one more time from the point of view,
discrete calculus. And what I'd like to emphasize
is a particular point of unity between discrete and continuous calculus. This is Newton's formula, something that
I've been saving for a special occasion. Let's begin. Recall that the forward
difference operator, delta, can be expressed as the shift,
E, minus the identity, I. If we rearrange terms a little bit,
then we can express the left shift, E, as delta + I. Now let's say that we want to
shift over not one slot, but n places, then we can express this
operator as a power, E to the n, which thus can be expressed as quantity,
delta + I, to the n. Now what happens when
we multiply that out? Well, we can use a binomial
theorem to express E to the n. As the sum, k goes from 0 to n
of the coefficient n choose k times delta to the k times I to the n- k. These binomial coefficients you recall
come from say Pascal's triangle or the definition in terms of factorials of
n and k and n- k and something like that. Anyhow, let's look at it
a little bit more carefully. Recall that the identity operator,
I, is the do nothing operator, so
we can effectively ignore it and just look at powers of
the forward difference, delta. Now, remember one of the useful
formulae from the script calculus is the notion of a falling power and
the falling k. It's n(n-1)(n-2) all the way to (n-k+1). But this is related to the binomial
coefficient as follows, n to the falling k can be expressed
as n choose k, times k factorial. Now that means that we can rewrite k to
the n as the sum of k goes from zero to n, as n to the falling k, divided by
k factorial, times delta to the k. And at this point, if you've been
paying attention in this class, you see something that
looks slightly familiar. If we take a sequence, a,
and consider the nth term, a sub n, this is really the shift. E to the n, applied to a,
evaluated at 0 at the initial slot. That means that we can
express a sub N as the sum. K goes from 0 to n of one over K factorial
times the Kth forward difference, delta to the K of a. Evaluated at 0 times,
and to the following k. This is Newton's formula in a slightly
different language than you'll find it in most books, but
in the language of discreet calculus, a language that makes you see immediately
the relationship to Taylor Series. Indeed Newtons formula is
the discreet calculus version of our classical formula for
the Taylor Series of f at x. Let's look at the similarities,
in both cases we have one of our k factorial and we have,
in the discrete version, the case forward difference
of a evaluated at 0. In the smooth version, we have the Kth
derivative of f evaluated at 0. These look a little bit closer,
if we replace that Kth derivative of f by the Kth power
of the differentiation operator D applied to f and then evaluated at 0. Whereas in the smooth
case we have x to the k. In the discrete case we have
the discrete version of this monomial. That is N to the falling k. This beautiful and fearful symmetry
between the notion of a Taylor Series for discrete and continuous calculus
is one of the gems in this course. If you don't mind,
I would like to make a few observations. Here at the very end. Recall the shift operator, E,
applied to a sequence, a, and evaluated at the nth term is
really just a sub n plus one. And in operator language,
it can be expressed as I plus delta. There's a corresponding shift operator
in the continuous world as well. E applied to a function f, and evaluated at x, is simply f at x + 1. Now, way back when, we argued using a Taylor series that
the shift operator in the smooth world is really the exponentiation of
the differentiation operator. That is, E is I plus D plus one over two factorial D squared
plus one over three factorial D cubed. Etc. And seeing the relationship between
the smooth and the discrete versions of a shift lead us to
a few interesting observations. First of all, from the observation that
the shift operator is the exponentiation of the differentiation operator we
can express the discreet derivative. The forward difference,
delta, as e to the d minus i. This gives us an explicit
connection between the notions of differentiation in the discrete and
the continuous world. Lastly, we could say that
differentiation D is really the natural log of the shift operator or
if you like the natural log of the identity plus
the forward discrete difference delta. Those are all beautiful observations and all are connected to this simple
initial question with which we began. What is e to the x? By now, you have seen so many answers to that question from so
many different perspectives. But you haven't seen it all. Indeed there is much more to
mathematics than just calculus. Even the type of calculus that
you have seen in this course. I hope that this course has been for
you more than a bag of tricks or formulae written into boxes. I hope that you have the seen the beauty,
and the wonder, that is implicit, both, in the world around us and
the mathematics that we use to model it. I hope that this course has been for you more than just a few video lectures
in which to up your skill set. Pay attention to the things around you. Go deeper. Maybe take a few more math classes
to see more of what's out there. It has been my privilege and
pleasure to be your professor this term.