Q 46. - Calculus

Question

Sketch the level curves $c = -3, -2, -1, 0, 1, 2, 3$ if they exist for the specified function. $f (x, y) = \sqrt{xy}$

Step-by-Step Solution

Answer

These are all quadratic equations. Hence, they all represent parabolas.

Step 1: Given information

The given function is $f(x,y)=\sqrt{xy}$

Step 2: Calculation

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 ${\mathrm{ℝ}}^2$

This function's graph will be in the ${\mathrm{ℝ}}^3$ 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.

$$\begin{gathered} \sqrt{xy}=-3\Rightarrow y=\frac{9}{x} \\ \sqrt{xy}=-2\Rightarrow y=\frac{4}{x} \\ \sqrt{xy}=-1\Rightarrow y=\frac{1}{x} \\ \sqrt{xy}=0\Rightarrow y=0 \\ \sqrt{xy}=1\Rightarrow y=\frac{1}{x} \\ \sqrt{xy}=-2\Rightarrow y=\frac{4}{x} \\ \sqrt{xy}=3\Rightarrow y=\frac{9}{x} \end{gathered}$$

Step 3: Explanation

These are all quadratic equations. Hence, they all represent parabolas.

Plot these equations on same $x y-$plane.