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
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