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\textbf{Partial Differentiation}

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

For the given families of level curves ($f(x,y)=C$) describe the
associated graphs of the function $f(x,y)$.

(It can be assumed that families correspond to values of $C$ that are
equally spaced. The behaviour of the given family is representative of
all families of the function.)

\begin{description}
\item{(a)}
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\item{(b)}
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\item{(c)}
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\item{(d)}
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\textbf{Answer}

\begin{description}
\item{(a)}
The graph is a plate containing the $y$-axis, sloping uphill towards
the right. It is similar to a function of the form $f(x,y)=y$.

\item{(b)}
The graph is a cylinder parallel to the $x$-axis, rising from zero
height, steeply to begin with, but more and more slowly as $y$
increases. It is similar to a function of the form $f(x,y)=\sqrt{y+5}$.

\item{(c)}
The graph is an inverted circular cone with its vertex at height 5 on
the $z$-axis and base circle in the $xy$-plane. It is similar to a
function of the form $f(x,y)=5-\sqrt{x^2+y^2}$.

\item{(d)}
The graph is a cylinder (possible parabolic) with its axis in the
$yz$-plane, and sloping upwards in the direction of increasing $y$. It
is similar to a function of the form $f(x,y)=y-x^2$.
\end{description}

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