Fundamental Theorem of Calculus - Proof of The First Part

Proof of The First Part

For a given f(t), define the function F(x) as

For any two numbers x1 and x1 + Δx in, we have

and

Subtracting the two equations gives

It can be shown that

(The sum of the areas of two adjacent regions is equal to the area of both regions combined.)

Manipulating this equation gives

Substituting the above into (1) results in

According to the mean value theorem for integration, there exists a c in such that

Substituting the above into (2) we get

Dividing both sides by Δx gives

Notice that the expression on the left side of the equation is Newton's difference quotient for F at x1.

Take the limit as Δx → 0 on both sides of the equation.

The expression on the left side of the equation is the definition of the derivative of F at x1.

To find the other limit, we will use the squeeze theorem. The number c is in the interval, so x1 ≤ c ≤ x1 + Δx.

Also, and

Therefore, according to the squeeze theorem,

Substituting into (3), we get

The function f is continuous at c, so the limit can be taken inside the function. Therefore, we get

which completes the proof.

Read more about this topic:  Fundamental Theorem Of Calculus

Famous quotes containing the words proof of the, proof of, proof and/or part:

    “If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
    —Polly Berrien Berends (20th century)

    “The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
    —Andrew Michael Ramsay (1686–1743)

    “The insatiable thirst for everything which lies beyond, and which life reveals, is the most living proof of our immortality.”
    —Charles Baudelaire (1821–1867)

    “When this immediate evil power has been defeated, we shall not yet have won the long battle with the elemental barbarities. Another Hitler, it may be an invisible adversary, will attempt, again, and yet again, to destroy our frail civilization. Is it true, I wonder, that the only way to escape a war is to be in it? When one is a part of an actuality does the imagination find a release?”
    —Ellen Glasgow (1873–1945)