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Math Help - Graph Transformation Problems

  1. #1
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    Graph Transformation Problems

    1. If a curve on a graph of f is F9x)=x^2-2x for all real numbers x, is the absolute value (f(x)) diffentiable at x=0? I'm pretty sure its no, but i'm not sure how to explain it.

    2. If the same curve exists, is the f(absolute value(x)) differentiable at x=0? Once again, i believe its no, but i can't explain it well.

    Thanks a lot

    -aussiekid90
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  2. #2
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    Quote Originally Posted by aussiekid90 View Post
    1. If a curve on a graph of f is F9x)=x^2-2x for all real numbers x, is the absolute value (f(x)) diffentiable at x=0? I'm pretty sure its no, but i'm not sure how to explain it.

    2. If the same curve exists, is the f(absolute value(x)) differentiable at x=0? Once again, i believe its no, but i can't explain it well.
    Consider the general absolute value y=|x|.
    Is it differenciable at x=0?
    No.

    Because, at x=0 we have.
    \lim_{\Delta x\to 0}\frac{|\Delta x|}{\Delta x}
    If we approach from the right we have 1.
    If we approach from the left we have -1.
    The sided limits do not conincide.
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  3. #3
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    G'day, aussiekid90!

    Your answers are correct . . .


    1. If f(x)\:=\:x^2-2x, is |f(x)| differentiable at x=0?
    The graph of f(x) \:=\:x^2-2x looks like this:
    Code:
                * |               *
                  |
                 *|              *
          --------*-------------*--
                 0| *         * 2
                  |    * *
                  |
    In the graph of g(x) \:=\:|x^2-2x|,
    . . anything below the x-axis is reflected upward.
    Code:
                * |               *
                  |    * *
                 *| *       *  *
          --------*-----------*--
                  |
    The graph has two different slopes at x = 0.
    . . The function is not differentiable there.



    2. For the same curve, is f(|x|) differentiable at x=0?

    The graph of f(|x|) is symmetric to the y-axis.
    The graph looks like this:
    Code:
          *               |               *
                          |
           *              |              * 
          --*-------------*-------------*--
              *         * | *         *
                   * *    |    * *
                          |

    The graph has two different slopes at x=0.
    . . The function is not differentiable there.

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