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Math Help - Question about a function

  1. #1
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    Question about a function

    Assuming the conditions for the MVT hold for f:\left[ {a,a + h} \right] \to R, so that for some \theta  \in \left( {0,1} \right) we have

    f\left( {a + h} \right) - f\left( a \right) = hf'\left( {a + \theta h} \right).

    If we fix f and a for each non-zero h, how would you write \theta \left( h \right) for a corresponding value of theta?
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  2. #2
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    Quote Originally Posted by james89 View Post
    Assuming the conditions for the MVT hold for f:\left[ {a,a + h} \right] \to R, so that for some \theta  \in \left( {0,1} \right) we have

    f\left( {a + h} \right) - f\left( a \right) = hf'\left( {a + \theta h} \right).

    If we fix f and a for each non-zero h, how would you write \theta \left( h \right) for a corresponding value of theta?
    You can't. All the MVT tells you is that f(a+h)- f(z)= h f( \xi) but says nothing about where \xi is.
    Last edited by Plato; March 8th 2009 at 01:27 PM. Reason: Fix LaTeX
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    You can't. All the MVT tells you is that f(a+h)- f(z)= h f( ^xi) but says nothing about where \xi is.
    It is definitely possible to complete with the information given - otherwise I wouldn't have been asked it.
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  4. #4
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    Quote Originally Posted by james89 View Post
    It is definitely possible to complete with the information given - otherwise I wouldn't have been asked it.
    Maybe you should explain your question more clearly.
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    I'm not sure what else I can say beyond what I have already given. The 'R' in the function definition is the set of real numbers, and I have been asked "Fix f and a, and for each non-zero h write theta(h) for a corresponding value of theta."
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  6. #6
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    Quote Originally Posted by james89 View Post
    I'm not sure what else I can say beyond what I have already given. The 'R' in the function definition is the set of real numbers, and I have been asked "Fix f and a, and for each non-zero h write theta(h) for a corresponding value of theta."
    In that case, I must agree with Halls.
    You said that you know it is possible. How do you know that?
    It may be that whoever wrote the question has a complete answer in mind.
    If I were you, I would discuss this the source of the question.
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