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)

    “In the reproof of chance
    Lies the true proof of men.”
    —William Shakespeare (1564–1616)

    “There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “Your toddler is no longer a baby feeling himself as part of you, using you as his controller, facilitator, his mirror for himself and the world. But he is not yet a child either; ready to see you as a person in your own right and to take responsibility for himself and his own actions in relation to you.”
    —Penelope Leach (20th century)