Q 53 - 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.

$f\left(x,y,z\right)=x+2y+3z$.

Step-by-Step Solution

Answer

The level surfaces are the planes with equations

$x+2y+3z=c; c=-3,-2,-1,0,1,2,3$.

Step 1. Given information

We have given the following function :-

$f\left(x,y,z\right)=x+2y+3z$.

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 :-

$f\left(x,y,z\right)=x+2y+3z$

We know that the level surfaces are determined as :-

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

That is we have :-

$x+2y+3z=c$

We know that this is the equation of plane for any value of $c$.

So that we can conclude that the level surfaces are the planes with equation $x+2y+3z=c$ for   $c=-3,-2,-1,0,1,2,3$.