Q 56 - Calculus

Question

In Exercises $53-58$, determine the level surfaces $c=-3,-2,-1,0,1,2,3$ if they exist for the specified function.

$f\left(x,y,z\right)=\frac{x^2+y^2}{z}$.

Step-by-Step Solution

Answer

The level surfaces are determined by the equation $x^2+y^2=cz$. For $c=0$, $x^2+y^2=0$ is represents z-axis in 3-D plane.

Also the level surfaces are circular paraboloid with equation $x^2+y^2=cz; c=-3,-2,-1,1,2,3$.

Step 1. Given Information

We have given the following function :-

$f\left(x,y,z\right)=\frac{x^2+y^2}{z}$.

We have to determine level surfaces $c=-3,-2,-1,0,1,2,3$ for the given function.

Step 2. Determine level surfaces

The given function is :-

$f\left(x,y,z\right)=\frac{x^2+y^2}{z}$.

We know that the level surfaces are determined as :-

$f\left(x,y,z\right)=c$.

That is we have :-

$$\begin{gathered} \frac{x^2+y^2}{z}=c \\ \Rightarrow x^2+y^2=cz \end{gathered}$$

For $c=0$, it becomes :-

$x^2+y^2=0$. It represents the z-axis in 3-D plane.

For all other values of $c=-3,-2,-1,1,2,3$, the level surfaces $x^2+y^2=cz$ are circular paraboloid.