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Math Help - expansion of function..?

  1. #1
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    expansion of function..?

    I've found the following statement in my textbook:

    If f: I \subset \mathbb{R} \rightarrow \mathbb{R} is a function of class C^2, than there exists \varepsilon \in <0, 1> such that f(x+h)=f(x) + h f'(x) + \frac{h^2}{2} f'' (x+\varepsilon h)

    This is not a consequence of anything we've done in the lesson, so I was hoping someone could tell me is what this expansion is.

    Thank you!

    (It reminds me of Taylor series, but it's not, I think..)
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  2. #2
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    Quote Originally Posted by marianne View Post
    I've found the following statement in my textbook:

    If f: I \subset \mathbb{R} \rightarrow \mathbb{R} is a function of class C^2, than there exists \varepsilon \in <0, 1> such that f(x+h)=f(x) + h f'(x) + \frac{h^2}{2} f'' (x+\varepsilon h)

    This is not a consequence of anything we've done in the lesson, so I was hoping someone could tell me is what this expansion is.

    Thank you!

    (It reminds me of Taylor series, but it's not, I think..)
    it is Taylor series indeed: f(b)=f(a)+(b-a)f'(a) + \frac{(b-a)^2}{2}f''(c), for some c between a and b. thus c= (1-\epsilon)a+\epsilon b, for some 0<\epsilon<1. now put b=x+h and a=x.
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  3. #3
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    Thank you!!!
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