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Math Help - need help integral theorems

  1. #1
    Junior Member
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    need help integral theorems

    State true and false
    1>If f and g are contineous and f(x)>= g(x) for a<x<b, then integral(a to b)f(x)dx>=integral(a tob)g(x)dx
    2>If f and g are contineous and f(x)>= g(x) for a<x<b, then f '(x)>=g'(x) for a<x<b
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  2. #2
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    #1 is a theorem. You should see if you could prove it. It is actually very important.


    #2: On [0,1] consider f(x) = 2 - x^2 \quad \& \quad g(x) = x^2 .
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  3. #3
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    Hello, Gracy!

    State true and false.

    (1) If f and g are continuous and f(x) \geq g(x) for a<x<b,
    then: . \int^b_a f(x)\,dx\:\geq \:\int^b_ag(x)\,dx

    I would say True . . .
    Code:
            |   *
            |   :*        :
            |   :  *      :
            |   :     *   :
            |   :         * f(x)
            |   :         :
            |   :         o g(x)
            |   :      o  :
            |   :  o      :
            |   o         :
          --+---+---------+--
            |   a         b

    If f(x) is always "above" g(x) on (a,\,b)
    . . then the area under f(x) will be \geq the area under g(x).



    (2) If f and g are continuous and f(x) \geq g(x) for a<x<b,
    then: .  f '(x) \geq g'(x) for a<x<b

    This is False . . .
    Code:
            |   *
            |   :*        :
            |   :  *      :
            |   :     *   :
            |   :         * f(x)
            |   :         :
            |   :         o g(x)
            |   :      o  :
            |   :  o      :
            |   o         :
          --+---+---------+--
            |   a         b

    While f(x) is above g(x), we see that
    . . f'(x) is negative while g'(x) is positive.

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