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Math Help - differentiable/continuous functions

  1. #1
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    May 2010
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    differentiable/continuous functions

    Determine all real values of a and b such that the function
    f(x) = ax+b for x =< 1
    = tan(Pi*x/4) for 1 < x < 2
    is differentiable at x = 1

    ?
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  2. #2
    Senior Member
    Joined
    Jan 2010
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    Is this the correct function?

    f(x) = \begin{cases} ax+b, & \mbox{if } x \le 1 \\ \tan ( \pi x/4 ), & \mbox{if } 1<x<2 \end{cases}

    You basically need to check for two things:

    1) The function must be equal on both sides as x \to 1
    2) The function's derivative must be equal on both sides as x \to 1

    For the first condition, plug in x=1 into both parts to get:

    a+b = \tan(\pi / 4)

    And the second condition, take derivatives of both sides, then plug in x=1 to get:

    a = \frac{\pi}{4} \sec^2 (\pi / 4)

    Once you have established these two equalities, you can now determine what a,b are.
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