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Let's say I have some function f,

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that is continuous on an interval between a and b,

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and I have these brackets here, so it also includes a and b and the interval.

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So let me graph this, just so we get a sense of what I'm talking about

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So that's my vertical axis, this is my horizontal axis.
I'm gonna label my horizontal axis t,

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so that we can save x for later,
and I can still make this y right over there

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and let me graph, this right over here is the graph of:
y is equal to f(t)

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Now, our lower endpoint is a, so that's a,
right over there

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Our upper boundary is b,

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...to make that clear, and actually just to show that we're including that endpoint,

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let me make them bold lines, filled-in lines.

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So lower boundary a, upper boundary b,
we're just saying, and I've drawn it this way,

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that f is continuous on that.

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Now, let's define some new function that's the area under the curve

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between a and some point that's in our interval.

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Let me pick this right over here, x.

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So let's define some new function, to capture the area under the curve,

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between a and x.

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Well, how do we denote the area under the curve between two endpoints?

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Well, we just use our definite intergral, that's our Riemann integral, that's really..

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that right now, before we come up with the conclusion of this video,

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it really just represents the area under the curve between two endpoints.

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So this right over here, we can say is,

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the definite integral from a to x, of f(t), dt.

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Now, this right over here is going to be a function of x,
let me make it clear,

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where x is in the interval between a and b. 
This thing right over here

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is going to be another function of x.
This value is going to depend

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on what x we actually choose.

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So let's define this, as a function of x,
so I'm gonna say that this is equal to

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uppercase F(x).

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So all fair and good, uppercase F(x) is a function,
if you give me an x value

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that's between a and b, 
it'll tell you the area under lowercase f(t)

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between a and x.

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Now, the cool part, the fundamental theorem of calculus,

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the fundamental theorem of calculus tells us,
let me write this down,

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it's a big deal. 
Fundamental theorem of calculus tells us,

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that if we were to take the derivative of our capital F,

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so if I were to take the derivative
of capital F with respect to x,

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which is the same thing as taking the derivative of this,
with respect to x,

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which is equal to the derivative of all of this business,
let me copy this

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So, copy, and then paste.
Which is the same thing,

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I've defined capital F as this stuff,
so if I'm taking the derivative of the left hand side,

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it's the same thing as taking the derivative of the right hand side.

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The fundamental theorem of calculus tells us,

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that this is going to be equal to,

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it's going to be equal to f,
lowercase f(x)

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Now, why is this a big deal?
Why does it get such an important title,

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as the fundamental theorem of calculus?

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Well, it tells us, that for any continuous function f,

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If I define a function, that is the area under the curve
between a and x right over here,

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that the derivative of that function is going to be f.

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So let me make it clear,

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every continuous f has an antiderivative F(x).

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That by itself is a cool thing,
but the other really cool thing,

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or I guess these are somewhat related, is, 
remember, coming into this,

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all we did is, we just viewed the definite integral
as symbolizing the area under the curve

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between two points.
That's where that Riemann definition

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of integration comes from. But now we see a connection,

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between that and derivatives.
When you're taking the definite integral,

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one way of thinking, especially if you're taking the definite integral between a lower boundary and an x,

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one way of thinking about it is that you're essentially taking an antiderivative.

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So we now see a connection,

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this is why it is the fundamental theorem of calculus,

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it connects differential calculus and integral calculus.
Connection between derivatives,

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and (maybe I should say antiderivatives), 
derivatives and integration,

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which before this video, we just viewed integration as area under the curve.

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Now we see it has a connection to derivatives.

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So how would you actually use the fundamental theorem of calculus, maybe in the context of a calculus class,

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and we'll do the intuition for why this happens,

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or why this is true, and maybe a proof in later videos,

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but how would you actually apply this right over here?

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Well, let's say someone told you, that they want to find the derivative,

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Let's say someone wanted to find the derivative with respect to x

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of the integral from, I don't know, I'll pick some random number here,

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Pi to x, of,
I'll put something crazy here,

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cosine squared of t, over the natural log of t minus the square root of t, dt.

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So they want you to take the derivative with respect to x of this crazy thing,

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Remember, this thing in the parentheses, is a function of x.

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Its value, it's going to have a value that is dependent on x,
you give it a different x,

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and it's going to have a different value.
So what's the derivative of this with respect to x?

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Well, the fundamental theorem of calculus tells us it can be very simple.

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We essentially, and you can even pattern match up here, 
we'll get more intuition of why this is true in future videos,

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but essentially, everywhere where you see this right over here is an f(t)

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everywhere you see a t, replace it with an x, and it becomes an f(x).

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So this is going to be equal to

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cosine squared of x, over the natural log of x minus the square root of x,

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you take the derivative of the indefinite integral, where the upper boundary is x right over here,

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just becomes, whatever you were taking the integral of, that as a function of, instead of t,

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that is now a function of x.

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So it can really simplify sometimes, taking a derivative,

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and sometimes, you'll see on exams,

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these trick problems where you have this really hairy thing that you need to take a definite integral of

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and then take the derivative,

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you just have to remember the fundamental theorem of calculus,

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the thing that ties it all together,
it connects derivatives and integration,

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you can just simplify it, by realizing that this is just going to be,

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instead of a function lowercase f(t), it's going to lowercase f(x),

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let me make it clear, in this example right over here,

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this right over here was lowercase f(t),

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and now it became lowercase f(x).

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This right over here was our a,

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and notice, it doesn't matter what the lower boundary of a actually is.

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You don't have something on the right hand side that is in some way dependent on a.

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Anyway, hope you enjoyed that and in the next few videos, we'll think about the intuition,

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and do more examples making use of the fundamental theorem of calculus.