[MUSIC]. 
I want to begin with a statement of he 
fundamental theorem of calculus. 
Suppose the function little f, from the 
closed interval a to b, to the real 
numbers, is a continuous function. 
And let's define a function big F, to be 
the accumulation function. 
Right, that's the integral. 
From a to x of the function on the left 
well then the function big F turns out to 
be continuous in the closed interval a to 
b differentiable on the open interval a 
to b and this is the big point the 
derivative of big F is little f. 
We've seen a bit of evidence for this 
already. 
And suppose I got the accumulation 
function. 
So big F of x is the integral from a to x 
of f of tdt. 
And now I can ask, how is the 
accumulation function changing? 
Well, the accumulation function is 
increasing, meaning it's derivative is 
positive if the function that I'm 
integrating is positive. 
And the accumulation function is 
decreasing, meaning the derivative is 
negative. 
If the function that I'm integrating is 
negative. 
And this suggests that the derivative of 
the accumulation function and the 
function that I'm integrating might be 
related in some way. 
Here's a graph that's particularly 
compelling. 
So, I'll draw the coordinate plane, 
here's the y axis, here's the t axis in 
this case and I'll draw some Random 
looking graph. 
That'll be the graph y equals f of t. 
And then I'll pick some point. 
I'll call it a and some other point x. 
And then I'll look at the area under the 
graph between a and x, rigth? 
And that's the integral from a to x of f 
of t d t. 
Right that area is the accumulation 
function evaluated x. 
Now what's the derivative of the 
accumulation function. 
What got the accumulation function, right 
f(x) equals the interval from a to x that 
f(t)dt and I don't know how that changes 
when x changes. 
So lets make a small change in x. 
Lets go from x. 
To say x plus h and then ask how does the 
area in this region compare to the area 
in this much larger region. 
What's big F of x plus h minus big F of 
x? 
Yeah, so what's F of x plus h right 
that's the area. 
From a to x plus h, minus F of x, big F 
of x. 
That's the area from a to x, and that 
difference, right. 
That difference is just the area inside 
here. 
That little leftover piece is practically 
a rectangle of height, f of x, little f 
of x and width h. 
I mean, this region is not quite a 
rectangle. 
But it's awfully close to a rectangle of, 
width h. 
And area f of x plus h minus f of x. 
Now, remember, by goal was to 
differentiate the accumulation function. 
And all I've got is this rectangle. 
Whose width is about h, and whose height, 
you think about it, it's about f of x. 
So what's up with this rectangle? 
Alright? 
I mean what I've really got here is a 
rectangle and its area is about f of x 
plus h minus f of x, this accumulation 
function. 
It's width is h, and it's height is about 
f of x. 
Well I've got a rectangle with this area 
and this width and, and this height. 
Alright, what do I know? 
I know its height times its width is its 
area, so if I take its area and divide by 
its width, I should get its height. 
Now all this is just approximate, but 
let's write that down. 
I The area of this you know, rectangle is 
f of x plus h minus f of x. 
And if I divide by the width, I should 
get, the height. 
What does that mean after I take a limit? 
Oh of course this is only approximately 
true, but if I take a limit, right. 
If I take a limit as h approaches 0. 
This gives me the derivative of the 
accumulation function and that will in 
fact be equal to the function's value. 
And that is the whole game. 
The accumulation function F is an 
antiderivative of f. 
Well this, I mean this is really 
remarkable. 
For instance you might be wondering 
what's an anti-derivative of this 
function, little f of x equals e to the 
negative x squared? 
Well if a functions continuous then we 
can find an anti-derivative. 
The anti-dirivitive is the accumulation 
function. 
So an anti-derivative of either the 
negative x squared is the accumulation 
function. 
I can define a function big f of x to be 
the intregal from a to x, for some number 
a, of e to the negative t squared d t. 
And if I differentiate this accumulation 
function, I get back e to the negative x 
squared. 
So I found an anti-derivative. 
For this function. 
Well I can't write it down any more 
nicely. 
Because I can't write down an 
anti-derivative for e to the negative x 
squared using the functions that I have 
at hand. 
But nevertheless, the anti-derivative is 
the accumulation function.