Intermediate Value Theorem

How to show that there are always two points opposite from each other with the same temperature on a circular wire ring? (這到底是數學題還是物理題?)

Update:

我想這一題應該不是用 IVT,而是用 勘根定理。 設 F(x)=f(x) - f(-x),若 F(a)F(b)<0,則必存在 F(c)=0,所以 f(c) = f(-c)。 f 可以定義為圓方程式之類的?

3 Answers

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  • 9 years ago
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    設溫度函數為T(θ), θ為ring上點的位置(角度),

    自然現象T(θ)為連續函數,且T(0)=T(2π)

    設F(θ)=T(π+θ)-T(θ),

    (1)F(θ)為 0~2π的連續函數

    (2)F(0)=T(π)-T(0)

    F(π)=T(2π)-T(π)=T(0)-T(π)

    F(0)*F(π) = - [T(π)-T(0)]^2 <=0

    若T(π)=T(0), 則θ=0, θ=π(相對兩點), 溫度相等

    若T(π) T(0), 則F(0)*F(π) <0, 則由IVT知: 必有F(k)=0, k介於 0~π

    則T(π+k)=T(k), 亦即角度k處的溫度與角度π+k處的溫度相等

    而k與π+k為ring上相對兩點, 故得證

    這是數學(微積分),不是物理!

    2011-05-23 07:41:34 補充:

    若T(π)≠ T(0), 則F(0)*F(π) < 0, 則由IVT知: 必有F(k)=0, k介於 0~π

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  • 9 years ago

    將溫度改為角度的函數,設f(x)=T(π+x)-T(x)即得

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  • linch
    Lv 7
    9 years ago

    勘根定理即是IVT的特例!!!

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