Question #138615
Sketch the level curves f−1(c) for the following functions:
a) f(x ,y ,z) = x−y^2 −z^2, c = −1,0,1.
1
Expert's answer
2020-10-18T17:55:18-0400

Given,

f(x,y,z)=xy2z2f(x,y,z)=x-y^2-z^2

Now, level curve is defined as

Gf={(x,y,z)R3f(x,y,z)=c}G_f=\{(x,y,z)\in \mathbb{R}^3|f(x,y,z)=c\}

That is, here, f1(c)=Gff^{-1}(c)=G_f

Now, set

f(x,y,z)=c    (xc)=y2+z2f(x,y,z)=c\implies (x-c)=y^2+z^2

Clearly, the level curves are family of paraboloid which passes through the center (c,0,0)(c,0,0)

Here, blue, green and red are level curve of f1(1),f1(0)&f1(1)f^{-1}(-1),f^{-1}(0)\& f^{-1}(1) respectively.


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