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Math Help - Intersection of two functions

  1. #1
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    Intersection of two functions

    y=cos(x) intersects y=x once,

    find A such that y=Acos(x) intersects y=x exactly twice.
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  2. #2
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    Hello gwtkof

    Welcome to Math Help Forum!
    Quote Originally Posted by gwtkof View Post
    y=cos(x) intersects y=x once,

    find A such that y=Acos(x) intersects y=x exactly twice.
    I think you will need to use a numerical/graphical method to solve this.

    The second point of intersection will be when the line is a tangent to the curve between x=0 and x=-\pi.

    Using an Excel spreadsheet, I set up graphs of the two functions, and found, by a series of approximations, this to be when x \approx -2.8, with A\approx 2.972.

    Grandad

    PS
    A more analytical approach is to say that the gradient of A\cos x at the point when it touches the line will be \displaystyle 1. So if you solve the equations
    A\cos x=x and -A\sin x = 1
    simultaneously, you'll end up with
    \tan x = -\dfrac1x
    which will again require a numerical method.

    This gives the same approximate values as I stated above.
    Last edited by Grandad; June 30th 2010 at 12:23 AM.
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  3. #3
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    Quote Originally Posted by Grandad View Post
    Hello gwtkof

    Welcome to Math Help Forum!
    I think you will need to use a numerical/graphical method to solve this.

    The second point of intersection will be when the line is a tangent to the curve between x=0 and x=-\pi.

    Using an Excel spreadsheet, I set up graphs of the two functions, and found, by a series of approximations, this to be when x \approx  -2.8, with A\approx 2.972.

    Grandad
    I think there's a more straightforward way. Let a be the value that you approximated as -2.8. The slope of the tangent of y=cos(x) at this point is -sin(a). The line going through the origin and the point of tangency is y=-sin(a)x. The point of intersection of the line y=-sin(a)x and the curve y=cos(x) gives us the relation

    cos(a) = -sin(a)a

    a = -cot(a)

    Then we can approximate a accordingly.

    Note: When I made this post it was just in response to the quoted portion, as the PS hadn't been posted yet.
    Last edited by undefined; June 30th 2010 at 12:44 AM.
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  4. #4
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    Thanks guys that helped alot.
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