Level Curves

  1. hi... I am new to this topic and frustrated.

    I have a curve f(x,y)= -3y/(x2 +y2 + 1)

    I was asked to draw a level curve of this and I'm not getting anywhere with it. If anyone has any pointers, or can help me with solving this question I would be gretfull. The only other thing this question asks is to describe it at the orgin or at (0,3) ( which is steeper).


    thanks for any help.
     
  2. jcsd
  3. Level curves are curves where the function is constant, and equals lets say to a number c.

    So you get an equation

    [tex]\frac{-3y}{x^{2}+y^{2}+1}=c[/tex]

    Rearranging this gives you

    [tex]cx^{2}+cy^{2}-3y=-c[/tex]
    [tex]cx^{2}+cy^{2}-2c\frac{3}{2c}y=-c[/tex]
    [tex]cx^{2}+cy^{2}-2c\frac{3}{2c}y+\frac{9}{4c^{2}}=\frac{9}{4c^{2}}-c[/tex]
    [tex]cx^{2}+(cy-\frac{3}{2c})^{2}=\frac{9}{4c^{2}}-c[/tex]

    Now you just have to investigate different values of c.
     
  4. HallsofIvy

    HallsofIvy 40,769
    Staff Emeritus
    Science Advisor

    That should be [itex]c(y- 3/(2c))^2[/itex]. That is, that leading "c" should be outside the parentheses.

     
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