91影视

The given function is $f(x,y)=4-\left(x^2+y^2\right)$

The goal is to draw the level curves for$c=-3,-2,-1,0,1,2,3$

Consider the function $f(x,y)$in two variables. This function is found in ${\mathrm{鈩�}}^2$

The graph of this function will be in ${\mathrm{鈩�}}^3$Assume the third variable to be $z$The equation of the graph is given as

$z=f(x,y)$

The graphs of this function's level curves are the graphs of the function for a constant value of $z$

Assume $z=c$The level curve of the graph at $c$is defined as the graph of the equation $f(x,y)=c$

Determine the equations for the various required level curves using the preceding definition.

$$\begin{gathered} 4-\left(x^2+y^2\right)=-3鈬�x^2+y^2=7 \\ 4-\left(x^2+y^2\right)=-2鈬�x^2+y^2=6 \\ 4-\left(x^2+y^2\right)=-1鈬�x^2+y^2=5 \\ 4-\left(x^2+y^2\right)=0鈬�x^2+y^2=4 \\ 4-\left(x^2+y^2\right)=1鈬�x^2+y^2=3 \\ 4-\left(x^2+y^2\right)=2鈬�x^2+y^2=2 \\ 4-\left(x^2+y^2\right)=3鈬�x^2+y^2=1 \end{gathered}$$

These are all equations of circles. Plot these equations on the same $xy-$plane.