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\textbf{Partial Differentiation}

\textit{\textbf{Functions of more than one variable}}
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\textbf{Question}

If $y=(x-C)^2$, are the curves level curves of a function $f(x,y)$?

To be the family of level curves of a function, what property must a
family of curves have in a given region of the $xy$-plane?


\textbf{Answer}

The curves $y=(x-C)^2$ are all horizontally shifted versions of the
parabola $y=x^2$, and they all lie in the half plane $y\ge 0$. Since
each of these curves intersects all of the others, they cannot be
level curves of a function $f(x,y)$ defined in $y \ge 0$. to be a
family of level curves, the curves must not intersect in the given
region.

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