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Math Help - Domain of a second derivative

  1. #1
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    Domain of a second derivative

    Hi everyone,

    I'm in Calc BC, but this year my teacher seems to make a lot of mistakes. I just wanted to ask a clarifying question about a second derivative of the function below:

    \\f(x) = \frac{x^8}{x^4} <br />
\\f^{''}(x) = ?

    I assumed like many of my classmates that we could just do this:

    \\f(x) = \frac{x^8}{x^4} = x^{4} <br />
\\f^{'}(x) = 4x^3 <br />
\\ f^{''}(x) = 12x^2

    But she said that we had to take into account the original function, and that since there was the x^4 as the denominator, we had to include the extra domain restriction of x \neq 0. This doesn't really make sense to me, because I figure you can just reduce it and that shouldn't be a problem. Obviously if the bottom was x^4-1 I would've included the restriction x \neq 1.

    Am I right to think this?

    Thanks,

    -
    Adam
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  2. #2
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    Ok based on elemental's response, I will go back to my original answer question I proposed to make you think about it: How can you have a derivative at point that DNE?
    Last edited by dwsmith; December 1st 2010 at 09:06 PM.
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  3. #3
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    No, you are not right to think this.

    If you have, for example, f(x) = \frac{x^2}{x}, x cannot equal 0. You can't cancel out the denominator when x is 0, the function is not continuous there.
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  4. #4
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    Why must we think of \frac{x^2}{x} and x as two different things? We were always taught that you can simplify the former into the latter. Is there any specific reason, or is it something I should just accept?
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  5. #5
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    Quote Originally Posted by aaddcc View Post
    Why must we think of \frac{x^2}{x} and x as two different things? We were always taught that you can simplify the former into the latter. Is there any specific reason, or is it something I should just accept?
    NO, you weren't taught that. You were taught that \frac{x^2}{x}= x for all x except x= 0. If you thought differently, you were not paying attention. You cannot divide by 0. The function f(x)= x has natural domain "all real x". The function f(x)= \frac{x^2}{x} has domain "all real x except x= 0".
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