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[MUSIC]. 

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Let's think about the fundamental theorem 
of calculus physically. 

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Let's define the function v by the rule 
that v of t is my velocity at time t. 

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Physically, what does the accumulation 
function of velocity mean? 

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Well that accumulation function, 
remember, is say, the integral from 0 to 

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b, of v of t dt. 
And this is really the distance, that 

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I've traveled from time 0 to time b. 
This also makes sense when we think back 

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to Riemann sums. 
But what would the Riemann sum say? 

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if I think about how far I traveled over 
a short time period, say, between time 

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zero and time h for some small number h, 
right? 

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How far did I travel during that time 
period? 

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Well, I traveled for that small time 
period h and I was traveling at a speed 

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of, I mean, you know, a velocity at time 
zero, say, is a good approximation. 

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And I imagine my velocity didn't change 
very much during this time period. 

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So, that's a pretty good approximation 
for how much I traveled during the first 

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h moments of my journey. 
What about between time h and time 2h, 

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right? 
How far did I travel there? 

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Well, the time that elapsed was h units 
of time. 

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And how fast was I going? 
Well, I could use v of h. 

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My velocity of time h as a good 
approximation for my velocity over that 

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time period, right? 
My velocity is not necessarily constant. 

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But this is standing in for a reasonable 
approximation and my velocity is not 

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changing too rapidly. 
All right. 

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This is how far I traveled, right? 
Time times velocity is distance. 

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All right? 
Now how far did I travel between, say 

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time 2h and time 3h. 
Well, again, how long was I traveling for 

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h units of time? 
And how fast was I going? 

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v 2h is a reasonable approximation for my 
velocity during that time period. 

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And of course this keeps on going, right? 
But what do I get if I add all of these 

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things up? 
Right? 

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What I'm getting is a Riemann sum. 
And if I keep on adding these things up 

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until I get, you know, all the way to, to 
b, right? 

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What I'm writing down is the 
approximation for this interval of a 

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particular Riemann sum which approximates 
this integral and in the limit as h goes 

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to zero, that Riemann sum will compute 
this integral. 

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So, summarizing that, the accumulation 
function of velocity, is displacement. 

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And what's the derivative of 
displacement? 

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It's velocity, right? 
The derivative of my accumulation 

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function in this specific case where the 
accumulation function is the accumulation 

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function for velocity, right, the 
derivative of that accumulation function 

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is the thing that I'm integrating, 
velocity. 

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That's the fundamental theorem of 
calculus. 

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Or in symbols, you know, the accumulation 
function, which is[INAUDIBLE] 

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displacement, is the interval from zero 
to b of my velocity. 

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And what I'm asking is, how is that 
changing, right? 

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If that's my displacement when I travel 
from times 0 to times b, right? 

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What is the derivative with respect to 
the time that I've been traveling? 

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The derivative of displacement is 
velocity. 

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So physically, differentiating the 
accumulation function of velocity is to 

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differentiate displacement. 
It's to ask for the rate of change in my 

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position, which is velocity, which is the 
integrand. 

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It's the fundamental theorem of calculus 
just wrapped up in physical clothing.