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Thread: Sketching level curves

  1. #1
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    Sketching level curves

    Sketch the level curves for the function $\displaystyle z = ln(x^2 + y^2)$.

    Attempt:

    $\displaystyle k = ln(x^2 + y^2)$

    $\displaystyle z = ln(k^2 + y^2)$

    $\displaystyle z = ln(x^2 + k^2)$

    Now I am not sure how I would draw each of these graphs due to the $\displaystyle ln(x^2 + y^2)$ part.

    Also when asked to sketch the level curves, does that mean you just let z = k and draw one graph? (as opposed to also letting x = k and y = k)
    Last edited by SyNtHeSiS; Dec 9th 2011 at 04:50 AM.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Sketching level curves

    Quote Originally Posted by SyNtHeSiS View Post
    Sketch the level curves for the function $\displaystyle z = ln(x^2 + y^2)$.

    Attempt:

    $\displaystyle k = ln(x^2 + y^2)$

    $\displaystyle z = ln(k^2 + y^2)$

    $\displaystyle z = ln(x^2 + k^2)$

    Now I am not sure how I would draw each of these graphs due to the $\displaystyle ln(x^2 + y^2)$ part.

    Also when asked to sketch the level curves, does that mean you just let z = k and draw one graph? (as opposed to also letting x = k and y = k)
    $\displaystyle z=k$

    $\displaystyle \ln (x^2+y^2)=k$

    $\displaystyle x^2+y^2=e^k$

    So, the level curves are circles in the xy-plane that are centered at the origin.
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    Re: Sketching level curves

    Thanks . Would drawing the graph give you a circular paraboloid (or do you have to draw the traces for $\displaystyle x = k $and $\displaystyle y = k,$ before drawing $\displaystyle x = ln(x^2 + y^2))$?
    Attached Thumbnails Attached Thumbnails Sketching level curves-circular-paraboloid.jpg  
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    MHF Contributor alexmahone's Avatar
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    Re: Sketching level curves

    Quote Originally Posted by SyNtHeSiS View Post
    Thanks . Would drawing the graph give you a circular paraboloid (or do you have to draw the traces for $\displaystyle x = k $and $\displaystyle y = k,$ before drawing $\displaystyle x = ln(x^2 + y^2))$?
    No, because the equation of a circular paraboloid is $\displaystyle x^2+y^2=a^2z$.
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    Re: Sketching level curves

    Oh ok. What steps would you use to draw the correct graph?
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  6. #6
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    Re: Sketching level curves

    Quote Originally Posted by SyNtHeSiS View Post
    Oh ok. What steps would you use to draw the correct graph?
    $\displaystyle z=\ln (x^2+y^2)=\ln r^2=2\ln r$

    $\displaystyle r=e^{z/2}$

    I have attached a graph generated using MATLAB.
    Attached Thumbnails Attached Thumbnails Sketching level curves-pic.png  
    Last edited by alexmahone; Dec 16th 2011 at 02:14 AM.
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  7. #7
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    Re: Sketching level curves

    Any function of the form $\displaystyle z= f(x^2+ y^2)= f(r)$ has circular symmetry. In particular, $\displaystyle z= ln(x^2+ y^2)= ln(r^2)= 2ln(r)$ is a logarithm graph. Draw that and rotate around the z-axis.
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