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Math Help - Level curves

  1. #1
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    Level curves

    I have to find some level curves for: f(x,y)=1-|x|-|y|

    So, if we call S at the surface given by the equation z=f(x,y), then

    z=1\Rightarrow{-|x|-|y|=0}\Rightarrow{x=y=0} \therefore P(0,0,1)\in{S}

    Now, that particular case its simple, cause it gives just a point, but if I go downwards I get:

    z=0\Rightarrow{-|x|-|y|=-1\Rightarrow{|x|+|y|=1\Rightarrow{|y|=1-|x|}}}

    I'm not sure how to represent this. How does this look on the xy plane?

    I know that:

    |y|=\begin{Bmatrix} 1-|x| & \mbox{ si }& y\geq{0}\\-1+|x| & \mbox{si}& y<0\end{matrix}

    And

    |x|=\begin{Bmatrix} x & \mbox{ si }& x\geq{0}\\-x & \mbox{si}& x<0\end{matrix}

    But it don't helps me to visualize the "curve". I know actually that it looks like a parallelogram, but thats because I've used mathematica to compute the surface :P I don't know how to deduce it analytically.
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  2. #2
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    Write y=\pm(1-|x|). Can you graph y=1-|x|?
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  3. #3
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    A level curve for f(x)= 1- |x|- |y| is given by 1- |x|- |y|= C, where C is a constant, or |x|+ |y|= 1- C.

    Now, it should be obvious that, since the left side of that is never negative, C cannot be larger than 1.

    If C= 1, the equation becomes |x|+ |y|= 0 which, since neither |x| nor |y| can be negative, is only true when x= y= 0. The level curve is the single point (0, 0).

    For C< 1, as always with absolute values, the simplest thing to do is to break the problem into cases.
    1. If x and y are both positive, |x|+ |y|= x+ y= 1- C. That is a straight line but remember to only draw it in the first quadrant.

    2. If x< 0 and y> 0 then |x|+ |y|= -x+ y= 1- C. Again, a portion of a straight line but now in the second quadrant.

    3. If x< 0 and y< 0 then |x|+ |y|= -x- y= 1- C. A portion of a straight line in the third quadrant.

    4. If x> 0 and y< 0 then |x|+ |y|= x- y= 1- C. A portion of a straight line in the fourth quadrant.

    If you draw those four segments for one value of C, say C= 0, it should be easy to see what the other level curves are.
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