Welcome to Calculus, I'm professor Ghrist. We're about to begin
Lecture 46 on Differences. >> We begin our construction
of a discreet calculus by building the notion of a derivative for
a sequence. This definition is going to turn out to be
much simpler than that of a derivative for smooth function. But this simplicity hides a profound depth that mirrors all the things we
have learned in calculus thus far. >> Recall that our goal is to
build a discrete calculus, a calculus based on sequences. In this lesson, we'll consider
what we mean by derivatives. This will come in the form of
differences or finite differences. The definition is as follows. Let's say,
that you have a sequence a sub n. We say, that the forward difference of a, delta a at n = a sub n+1- a sub n. That is, you look at the person
in front of you, and subtract off your present value. Now, if we compare this to
the definition of the derivative, we see that there are some commonalities. We are taking a difference in
the output of the function dividing by a difference in
the input of the function. In the case,
where that difference is equal to 1, because we're talking about a sequence. Now, if we were to try to graph this
function and give an interpretation in terms of slope, then by connecting
the dots, well, this would be the slope of the line segment in front
of you, hence the forward difference. This motivates the definition
of the backward difference. If you look behind you, and
consider the slope of at line, we define the backward difference,
nabla a at n to be a sub n- a sub n- 1. Let's consider a few examples. If we take the sequence 4n and compute it's forwards difference,
what do we obtain? I'm going to leave the verification
of the computation to you. I want you to show that one
obtains the constant sequence 4. Now, what do you observe here,
it is if we are taking the derivative of a linear function and
obtaining the constant function, but here these are sequences and
differences instead of derivatives. Here's another example a bit more
interesting the Fibonacci sequence. What happens when we take the forward
difference of that 1 finds, that again,
1 obtains the Fibonacci sequence, but shifted over by one step. With an additional 1 out in front,
that seems like it must be significant. For a last example,
consider the sequence 2 to the n. What happens when we compute
the forward difference of that? We obtain, again, 2 to the n. It says, if this is something, like the exponential function e to the x,
which is its own derivative, but here we're in the discrete
world taking differences. There are many parallels between
differences and derivatives. For example, if we ask the question,
which sequences are polynomial? Consider n squared,
when we take its forward difference, we obtain a sequence of odd numbers. That is the sequence, 2n + 1. What happens when we difference, again, the second forward difference delta
squared is in this case, a constant. The constant sequence 2, what happens when we take the next
difference, the third forward difference? Then we obtain the constant sequence 0. This is very similar to what
happens when we differentiate a polynomial after a finite
number of steps, we get 0. In general,
one can say that a sequence, a, is a polynomial of degree p if
the p plus first derivative, that is difference of a is the constant 0. Now, notice that there's
something that's not quite according to what you would expect here. That is the difference
of n squared is not 2n, but rather, 2n + 1. Now, that seems anomalous, but we can
explain that with a bit more notation, in particular, that of the falling powers. These are discrete calculus
versions of monomials. We say,
that n to the falling k is n times n- 1 times n- 2,
all the way down to n- k + 1. That works for a k bigger than 0. For k = 0, we'll define n to
the falling 0 to be 1, of course. Now, the reason why this is so useful is that the forward
difference of n to the falling k is k times
n to the falling k- 1. Let's look at this in the context
of the example that we've done. Consider the sequence n squared, we could rewrite n
squared as n(n- 1) + n), that is, it's really n to the falling
2 plus n to the falling 1. And hence, the forward difference of n
squared is the forward difference of n to the falling 2 + n to the falling 1. Differencing, like
differentiating is linear. Hence, this is the difference of n to the falling 2 + the difference
of n to the falling 1. That is 2n to the falling
1 + n to the falling 0. Otherwise written as the sequence 2n+1,
which is what we observed. And that seems a bit complicated,
why would you want to do that? Well, let's consider a more
fundamental question, the question that began this course. What is e? Well, we know that e is 1 +
1 + 1/2 + 1/6 + 1/24, etc. That is the sum k goes
from 0 to infinity of x to the k over k factorial
evaluated at x = 1. Now, that's for smooth calculus. What about discreet calculus? What is the discreet version of e? Let's try the same thing, but
using the discreet version of x to the k, that is n to the falling k. Consider the sum, k goes from 0 to infinity of n to
the falling k over k factorial. Evaluated that n = 1. What is that? That's 1 + n + 1/2 n(n- 1) + 1/6 n(n- 1)(n- 2), etc. What happens when we
evaluate this if n = 1? Most of the terms vanish. All of the terms vanish,
except the first two. When we evaluate at n = 1, we get 1 + 1,
which as you know is equal to 2. Therefore, in discrete calculus e
really takes on the value of 2. And in fact,
if we look at the version of e to the x, that is the sum of n to
the falling k over k factorial, we get precisely the sequence 2 to the n. And this is why, when we observed that the
forward difference of 2 to the n is 2 to the n, it reminded us of
the behavior of e to the x. At this point you may feel
a bit like Alice in Wonderland. What is with discrete e, and
2 to the n, and falling powers? Discrete calculus can be a strange place,
but there is a more rigorous approach
that you may find assuring. This approach uses
the perspective of operators. These are things that transform
functions into functions, or in this case, transform sequences. Some of these operators you've met before. For example, the identity operator. It acts on sequences the way you'd expect,
it does nothing. The identity applied to a,
that is Ia, is the same sequence, a. What's nice about operators is that
we can work with them algebraically. We can say that I squared,
that is applying the identity twice, is the same thing as applying
the identity once, it does nothing. E is the left shift operator,
that is Ea is the sequence, which in the nth slot,
gives you a sub n+1. And again, we can take powers of
E to shift further to the left. If there's a left shift,
then there must be a right shift. This has the notation E inverse,
it undoes E. Now, we can take powers
of this operator as well, and start multiplying them together. They behave in the way
that you would expect. E inverse E, gives you E to the 0,
that is the identity operator. I, that works as well, if you reverse
the order, it doesn't matter. A few other operators that we've
seen are the forward difference, and the backward difference. We can write each of these in
the language that we've already built. The forward difference, delta is E- I. The backward difference is I- E inverse. You want to have these two formulae
memorized, but why they're important? Here is an application
to higher differences. Let's say, you wanna compute the second
forward difference of a at n. This is the forward difference
of the forward difference of a. That is the forward difference
of a sub n + 1- a sub n, which with a little simplification gives a sub n + 2- 2a sub n + 1 + a sub n. What do you notice about
those coefficients? Well, if we rewrote
the difference operators, E- I and square that,
then using the algebra of operators, and simplifying according to what I does,
we obtain E squared- 2E + I, not simple. In more generality,
if we have the kth forward difference, then by multiplying out E- I k times,
we can obtain the formula. The sum is i goes from 0 to
k of negative 1 to the k- i times the binomial coefficient k choose i,
E to the i. Now, this means what? Let's say, that you wanted to
compute a high difference. Let's say, the eighth forward difference,
that's gonna depend on a lot of terms. What is the coefficient in
front of the a sub n + 6 term? Well, by writing down Pascal's triangle, looking up the appropriate coefficient
with the minus sign in the right place, we easily see that,
that coefficient is -56. That is much simpler than trying
to work it all out by hand. There are other things that we
seem to be able to do as well. Consider the discrete notion
of an indefinite integral. If the forward difference is E- I,
then what happens when we try to take the inverse,
that is undo differencing. Well, we need to take E- I
to the negative 1 power. Let's say, if I wrote that
a little bit differently and say put a minus sign out front,
and we call it I- E. Then it is as if we're trying to compute, 1 over I minus something. Well, I've seen formulae for 1 over
1- x in terms of a geometric series. What happens, if we tried applying
the geometric series to this formula? It would suggest that the inverse
operator to forward differencing is -(I + E + E squared + E cubed
+ E to the fourth, etc., etc. Now, that seems rather dubious. Let's check it on a simple example. See what we get. Let's take a random sequence. Let's say, 3, -1, 4, -1, 5, -9, 2, -6, 5, and I'm tired, so
I'm just gonna write 0s from now on. What would this delta inverse really mean,
it means what? Well, I need to at the nth term take the sum of all the terms
in a that follow, from n on up. And then put a minus sign in front of it. If we do so, walking down the line,
we can marching from the right, certainly compute
the terms in this sequence. This is easy,
since our sequence terminated in 0s. What happens when we take that,
and forward difference it? I'll leave it to you to check that
we obtain the original sequence a, that this actually does work. Now, don't get too carried away. You have to be careful. First of all,
[LAUGH] you forgot the constant. We all forgot the constant. Also notice that this is only
going to work if your sequence terminates in 0s, or
rather is sumable to the right. Have fun with discrete calculus,
but be careful. >> We've seen how finite differences
provide a discrete version of a derivative for sequences, complete with interesting
analogs of the exponential function, and a surprising way to do anti-differencing. In our next lesson, we're going to go
fully digital and use our notions of differencing and anti-differencing to give
an overview of the discrete calculus.