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- Hollywood
Dear All!
Proof ofthe second fundamental theorem
Please, how do we get in the sixth line that f(x) is right the one f(t), also, the sam f function as we have in the lines before?
In other words, how do we KNOW that The SAME f(x) we have in the right side before?
I understand that it is an DERIVATIVE, SURE! "An derivative! " But how do we know it is the same FUNCTION as we have f(t) on the right?
Many thanks!
It is the "mean value theorem". As long as F is differentiable, there exist c between x and x+ h such that (F(x+h)- F(x))/h= F'(c)= f(c) because F is defined as the anti-derivative of f. As h goes to 0, since c is always between x and x+h, c goes to x and f(c) becomes f(x).
Hello!
Thank U all very much, but I think we did not understand each other.
Sure it is clear that F'=f
My question is:
how do we know that this f from the left side of the eqation is the SAME f we have on the right side.
Proof ofthe second fundamental theorem
The functin on the right side we marked as "f".
Then, the derivative function on the left side we get we name AGAIN WITH THE SAME NAME! (f)
hOW can we know that f=f?
p.s. we could name the derivative on the left g8x) couln't we?
So, what woudl tell us then that g(x)=f(x)?
many thanks if someone has nerves for this-and sory if I am too intrusive!
Ah! "F" is defined as the area under the curve. In that case, we don't know that " ", it is NOT, in general, true. What is true is that where is some number between and . That is true because of the "intermediate value property".
If we were to draw a horizontal line at where is the maximum of f on the interval from x to , we get a rectangle, of area which completely contains the area under the graph: .
If we were to draw a horizontal line at where is the minimum of f on the interval from x to , we get a rectangle, of area which is completely contained in the area under the graph: .
(Using the intermediate value theorem, as well as guarenteeing that those "maximum" and "minimum" values exist requires that f be continuous. More generally, if f is not continuous at some points (but stays finite) we can work between points of discontinuity. The anti-derivative is the same as the 'area under the curve' (Riemann integral) as long as f has only a finite number of points of discontinuity and is bounded on the interval.)
Putting those together which, since , is the same as .
By the "intermediate value property then, there exist a value of x, , between and , such that .
It is taking the limit as goes to 0 so that goes to x, and is "trapped" between them, that gives F'(x)= f(x).