[MUSIC].
I've been trying to sell you on an analogy.
The analogy is that integration is a whole lot like summation.
Now we've also seen the fundamental theorem of calculus, and the fundamental
theorem of calculus tells us that integration practically amounts to
antidifferentiation. So how does the fundamental theorem of
calculus fit into this analogy? Well here's the analogy square.
I've got integrating and differentiating. And integrating is like summing.
So what's the thing that's like differentiating but for the discrete
world? I'm claiming it's differencing.
Now how so? Well, let's suppose that I've got a list
of numbers. Made my list of numbers is something easy
like one, two, three, four, five, it could be more creative.
But let's suppose this is my list of numbers.
The accumulation function just amounts to adding up those numbers.
So in this case, it'd be 0 plus 1, 1 plus 2, 3 plus 3, 6 plus 4, 10 plus 5, and so
forth. The difference operator, it's like
differentiation, but the difference operator amounts to writing out a new
list. The list of differences between success
of numbers. So in this case if I just had this list
of numbers, I could write down the differences between the subsequent
numbers in that list. 1 minus 0 is 1, 3 minus 1 is 2, six minus
3 is 3, 10 minus 6 is 4, 15 minus 10 is 5, and so forth.
Now if I were to look at the list of differences in the list of the
accumulation function, I'd get back the original list that I started with.
It's really worth repeating. Right?
Taking differences between the sum of the first k numbers in my original list and
the sum of the first k minus 1 numbers in my original list.
Just gives me back my original list of numbers.
And this is an analogous to the statement from calculus.
That if I take the derivative of the accumulation function, I give back the
original function. Mathematics isn't just about isolated
facts, It's really about analogies between ideas.
It's just like in literature where metaphor plays such an important role.