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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Preview Activity 11.8.1 .
(a) State the equation and shape of the level curves for the function
\(f(x,y)=x^2+y^2\) for the values
\(-2,-1,0,1,2\text{.}\)
Hint .
Some of these level sets will be empty, meaning that there are no
\((x,y)\) points that have the particular
\(k\) -value as output.
(b) State the equation and shape of the level curves for the function
\(g(x,y)=x^2-y^2\) for the values
\(-2,-1,0,1,2\text{.}\)
(c) For a constant
\(k\) and a function
\(f\text{,}\) the
level set is the set of points
\(P\) for which
\(f(P)=k\text{.}\) The remainder of this Preview Activity asks you to consider level sets of some functions of three variables. We use the term “level set” here because the set of inputs for a function of three or more variables that gives a particular output value will likely not be a curve.
State the equation and shape of the level sets for the function
\(f(x,y,z)=x^2+y^2+z^2\) for
\(-2,-1,0,1,2\text{.}\)
Hint .
Some of these level sets will be empty.
(d) State the equation and shape of the level sets for the function
\(g(x,y,z)=x^2+y^2-z^2\) for
\(-2,-1,0,1,2\text{.}\) These level set values should give three different types of surfaces.
(e) State the equation and shape of the level sets for the function
\(h(x,y,z)=x+y-z\) for
\(-2,-1,0,1,2\text{.}\)
