[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.