91影视

The given function is $f(x,y)=\frac{3x}{y}$

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

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

This function's graph will be in the ${\mathrm{鈩�}}^3$format. Assume that the third variable is $z$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.

$$\begin{gathered} \frac{3x}{y}=-3鈬�y=-x \\ \frac{3x}{y}=-2鈬�y=-\frac{3}{2}x \\ \frac{3x}{y}=-1鈬�y=-3x \\ \frac{3x}{y}=0鈬�x=0 \\ \frac{3x}{y}=1鈬�y=3x \\ \frac{3x}{y}=2鈬�y=\frac{3}{2}x \\ \frac{3x}{y}=3鈬�y=x \end{gathered}$$

These are all linear equations. Hence, they all represent straight lines. Plot these equations on the same $xy-$plane.