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[MUSIC]. 

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I've been trying to sell you on an 
analogy. 

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The analogy is that integration is a 
whole lot like summation. 

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Now we've also seen the fundamental 
theorem of calculus, and the fundamental 

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theorem of calculus tells us that 
integration practically amounts to 

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antidifferentiation. 
So how does the fundamental theorem of 

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calculus fit into this analogy? 
Well here's the analogy square. 

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I've got integrating and differentiating. 
And integrating is like summing. 

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So what's the thing that's like 
differentiating but for the discrete 

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world? 
I'm claiming it's differencing. 

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Now how so? 
Well, let's suppose that I've got a list 

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of numbers. 
Made my list of numbers is something easy 

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like one, two, three, four, five, it 
could be more creative. 

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But let's suppose this is my list of 
numbers. 

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The accumulation function just amounts to 
adding up those numbers. 

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So in this case, it'd be 0 plus 1, 1 plus 
2, 3 plus 3, 6 plus 4, 10 plus 5, and so 

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forth. 
The difference operator, it's like 

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differentiation, but the difference 
operator amounts to writing out a new 

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list. 
The list of differences between success 

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of numbers. 
So in this case if I just had this list 

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of numbers, I could write down the 
differences between the subsequent 

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numbers in that list. 
1 minus 0 is 1, 3 minus 1 is 2, six minus 

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3 is 3, 10 minus 6 is 4, 15 minus 10 is 
5, and so forth. 

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Now if I were to look at the list of 
differences in the list of the 

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accumulation function, I'd get back the 
original list that I started with. 

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It's really worth repeating. 
Right? 

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Taking differences between the sum of the 
first k numbers in my original list and 

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the sum of the first k minus 1 numbers in 
my original list. 

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Just gives me back my original list of 
numbers. 

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And this is an analogous to the statement 
from calculus. 

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That if I take the derivative of the 
accumulation function, I give back the 

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original function. 
Mathematics isn't just about isolated 

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facts, It's really about analogies 
between ideas. 

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It's just like in literature where 
metaphor plays such an important role.