Find the level curves for the function \(f(x,y)=x^2+y^2\) over the values \(k=\{-2,-1,0,1,2\}\text{.}\) State the equation and shape of the level curves for each value.
Find the level curves for the function \(g(x,y)=x^2-y^2\) over the values \(k=\{-2,-1,0,1,2\}\text{.}\) State the equation and shape of the level curves for each value.
In general, the level set of a function \(f\) will be the set of points for which \(f(P)=k\) for some constant \(k\text{.}\) We will look at some algebraically simple functions of three variables to see what kinds of level sets are possible. 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.
Find the level set for the function \(f(x,y,z)=x^2+y^2+z^2\) over the values \(k=\{-2,-1,0,1,2\}\text{.}\) Be sure to state the equation and shape of the level set for each value.
Find the level set for the function \(g(x,y,z)=x^2+y^2-z^2\) over the values \(k=\{-2,-1,0,1,2\}\text{.}\) Be sure to state the equation and shape of the level set for each value. These level set values should give three different types of surfaces.
Find the level set for the function \(h(x,y,z)=x+y-z\) over the values \(k=\{-2,-1,0,1,2\}\text{.}\) Be sure to state the equation and shape of the level set for each value.
