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Math Help - Smallest circle

  1. #1
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    Smallest circle

    The problem is as follows:
    Given the two functions f(x)= e^x and f(x)=ln(x),
    A.) Find the smallest possible circle that touches each function only once and all values of x and y are positive.
    B.) Find the relationship between x and the radius of the circle.

    Seeing how this seems like an optimization problem I wanted to see a calculus approach to this.
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  2. #2
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    I would consider the minimum distance between your functions and y=x,
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  3. #3
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    How would I do that if Im not given any point from which to begin?
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  4. #4
    Senior Member Tinyboss's Avatar
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    Since e^x and ln x are inverse functions, they're symmetric across the line x=y. So immediately you know that the center of your circle is on that line somewhere. Furthermore, you only need to minimize the distance of one of the functions to that line; the minimum distance of the other function will occur at the reflection of that point across x=y.

    To minimize the distance between e^x and x=y, the first idea that comes to mind is rotating your coordinate system 45 degrees clockwise. Then x=y becomes the x-axis, and you can use calculus to minimize whatever function e^x becomes.
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  5. #5
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    consider (e^x)-x. If you take its derivative, you get e^x-1. (e^x)-x has a stationary point (obvious from the graph that that is a minimum) when e^x-1=0, so at x=0. In order to find out where the circle is tangent to e^x, you could make e^x=-x, for example
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