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Math Help - integrals differening by konstant

  1. #1
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    integrals differening by konstant

    i think thsi is easy but cant think of the answer can some1 help me out
    thanx
    edgar

    Suppose f is continuous on a domain D. Prove that any two primitives
    of f (if they exist) differ by a constant.
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  2. #2
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    Quote Originally Posted by edgar davids View Post
    i think thsi is easy but cant think of the answer can some1 help me out
    thanx
    edgar

    Suppose f is continuous on a domain D. Prove that any two primitives
    of f (if they exist) differ by a constant.
    Primitive as in "anti-derivative"?
    ---
    Theorem: If a function f is differenciable with the property that f'(x)=0 for all points then, f(x)=C for some real number C.

    Proof: Let a,b and be be points a<b. Then, the closed interval [a,b] is continous (because differenciability implies continuity). And the open interval (a,b) is differenciable because that is the property of the function. This satisfies Lagrange's Mean Value theorem. There exists a point c\in (a,b) such as,
    f'(c)=\frac{f(b)-f(a)}{b-a}
    But, f'(c)=0
    Thus,
    \frac{f(b)-f(a)}{b-a}=0
    Thus,
    f(b)-f(a)=0 thus, f(a)=f(b). Which means that the evaluation of the function is invariant, i.e. it is a constant function. Q.E.D.

    Corrolarry
    Given a function f on some interval. If F and G are primitives, that is, F'=G'=f for all points in the open interval. Then, F'-G'=(F-G)'=0, thus, F-G is derivative zero throughtout the interval. By the theorem that means that, F-G=C for some constanct function C. Thus, F=G+C. Q.E.D.
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