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Math Help - Sketch z = arctan(x^2 + y^2)

  1. #1
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    Sketch z = arctan(x^2 + y^2)

    Sketch the level curves for z = arctan(x^2 + y^2). What does the graph of the function look like?

    Attempt:

    Let x = k:

    1. z = arctan(k^2 + y^2)

    Let y = k:

    2. z = arctan(x^2 + k^2)

    Let z = k:

    3. k = arctan(x^2 + y^2)

    Now what I am not sure of is how would you draw 1-3? Would you have to use all the steps for curve sketching e.g. first derivative, second derivative etc?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Take \tan in booth sides. For example in 3) : x^2+y^2=\tan k (circumference) .


    Fernando Revilla
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  3. #3
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    How would you plot tanz = x^2 + k^2 from 2)?
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    The level curves occur when the function

    a=f(x,y)=k a constant. This gives

    k=\tan^{-1}(x^2+y^2) As noted above this gives

    \tan(k)=x^2+y^2 This is the equation of a circle centered at the origin with radius r=\sqrt{\tan(k)}
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  5. #5
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    Oh so by level curves the question actually means you must just let z = k (not x = k and y = k) and hence draw that graph?
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