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.