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$f(x) = y^2 - x^2 y -2x^4 + \epsilon x$ , where $\epsilon>0$ and $\epsilon$ is small.

Draw a contour plot in $\mathbb{R^2}$of f(x).

How do I do this? I have no idea, it is not a simple circle.. Please, help

Edit: I forgot to tell, we CANNOT use computer programs. That is why I dont know how to start, I don't know methods for such task.

Martin Sleziak
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2 Answers2

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For $\epsilon = 0$, $f(x)$ factors as $(y - 2 x^2)(y + x^2)$. So the contour $f = 0$ would consist of two parabolas $y = 2 x^2$ and $y = -x^2$. Above the top parabola and below the bottom one $f$ would be positive, between them it would be negative. In particular, on the $x$ axis it would be $-2x^4$. Now how would adding a term $\epsilon x$ affect this?

Robert Israel
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In Mathematica:

Plot3D[y^2 - x^2 y - 2 x^4 + 1 x, {x, -4, 4}, {y, -4, 4}]

enter image description here

DumpsterDoofus
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    That is not a contour plot since your graph doesn't contain 2-D curves. Instead, it's an actual plot of the graph. – NasuSama Jan 12 '14 at 20:07
  • Sure, but the contour plot looks boring, and it's harder to make out what the function looks like in the contour plot because it's just a bunch of near-parallel lines. – DumpsterDoofus Jan 12 '14 at 20:08
  • @NasuSama: Moreover, the OP wanted us not to use any computer programs. This is really boring. – Mikasa Jan 12 '14 at 20:10
  • you are at risk of being down-voted for some cannibals that lurk – janmarqz Jan 12 '14 at 20:29