The Fundamental Theorem of Calculus (Part I)

The Fundamental Theorem of Calculus (Part I)

The Fundamental Theorem of Calculus says that if f is differentiable at a point x, then f'(x) = 0. In other words, the derivative of a function is zero at any point where it's not differentiable. The Fundamental Theorem of Calculus is used to prove that definite integrals have unique solutions. This theorem states that if f is differentiable on an interval [a, b], then for every number c in that interval, we can find a number d such that f(b) - f(a) = F(b - a) * d where F is the antiderivative of f.

We now come to a very important theorem in calculus. It's called the Fundamental Theorem of Calculus, and it has two parts:

In part one, if f is the continuous function defined on the closed interval [a, b], then we can integrate f over that interval. If we integrate from a to x where x lies between a and b, then this becomes a function depending on x.

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Now, the function x is continuous on the whole interval [a, b] and differentiable on the open interval [a, b].

In addition, if we differentiate this function with respect to x, we find that F'(x) = small f(x), where F(x) is the original function and ' indicates differentiation with respect to x.

This is part one. It says that you differentiate this definite integral with respect to x, the upper limit, and it gives you back the original integral.

Part two of the fundamental theorem of calculus states that if a function f is integrable on an interval a to b, then the definite integral of small f over that interval equals G(x) - G(a), where G(x) is any antiderivative of f on the interval. This is part b.

In this part two, the left-hand side is the definite integral while the right-hand side involves an indefinite integral or antiderivative. Therefore, this equation connects definite integrals with indefinite integrals.

One of the important consequences of this is that if we want to find the definite integral on the left-hand side, we can find it by finding any antiderivative of f and evaluating it at b and a.

Now, to find the definite integral of a function, we need not go through a very complicated and lengthy procedure in which we divide the interval [a, b] into thin rectangles and compute Riemann sums and take limit.

This complex and lengthy procedure can be computed by finding antiderivatives. This is important because it demonstrates that finding antiderivatives reduces the calculation of definite integrals to a simpler task.

Here, I will recall for you that the indefinite integral of f(x) without the limits and so forth is called an antiderivative of f. The indefinite integral denotes antiderivatives of f and is thus called an antiderivative or indefinite integral.

In this way, let’s explain the geometric meaning of the Fundamental Theorem of Calculus. In the figure above, the line in blue denotes f(x), and this function is integrated from a to x. The area under this graph from a to x is F(x). So this denotes the area under this graph from a to x.

If we move the point x to the next point , x + h here, we get F(x + h). This is the area from a to x + h. If we do subtraction, F(x + h) minus F(x) then we get now the area in this region.

So, it equals to the definite integral of f(x) from x to x plus h. Now, this area is close to the area of the rectangle. The rectangle has space equal to h and height f(x), so it is is

approximately equal to it but not exactly equal to it. There is a difference between the exact area and this area rectangle which is this little piece we call excess.

Now, excess is the difference between these two areas.

F(x+ h) - F(x) is equal to the area of the rectangle plus the excess there.

So, you divide everything by h. This gives you f(x). And this one divided by h is this piece. When you divide this by h, you get this piece.

As seen in the figure, the excess is no more than the area of this rectangular piece. The height of this rectangular piece is h, and its height is s.

Thus, the area of the excess is less than or equal to s times h when h tends to zero. Because the function is continuous, s also tends to zero as h approaches zero.

When h tends to zero, f(x) tends to this expression. This expression approaches zero when h is small enough. Moreover, this one goes to zero while this one goes to f(x). In other words, when h approaches zero, f(x) approaches its value at x=h.