Solving a Calc 3 Problem: Finding a Level Surface at (1,-2,0)

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SUMMARY

The problem presented involves finding the level surface of the function f(x,y,z) = xyz + 3 that passes through the point (1, -2, 0). The constant for the level surface is determined to be k = 3, leading to the equation xyz = 0. This equation represents three planes in the coordinate system, specifically the xy-plane, xz-plane, and yz-plane, which intersect at the origin. The discussion concludes that the equation xyz = 0 effectively defines the level surface, encompassing more than just the coordinate axes.

PREREQUISITES
  • Understanding of multivariable calculus concepts, specifically level surfaces.
  • Familiarity with the function notation and evaluation in three dimensions.
  • Knowledge of parametrization techniques in calculus.
  • Basic comprehension of geometric interpretations of equations in three-dimensional space.
NEXT STEPS
  • Explore the concept of level surfaces in multivariable calculus.
  • Learn about parametrization of surfaces in three dimensions.
  • Study the geometric interpretation of equations like xyz = 0.
  • Investigate the implications of level curves and surfaces in optimization problems.
USEFUL FOR

Students of multivariable calculus, educators teaching calculus concepts, and anyone interested in the geometric interpretation of functions in three-dimensional space.

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This is a problem I got from a Stanford class in calc 3:

Let f(x,y,z)=xyz+3. Find an equation of the level surface that passes through the point (1,-2,0).

This is as far as I have gotten:
The constant for the level surface will be k = xyz + 3 = (1)(-2)(0) + 3 = 3.
The equation is thus 3 = xyz + 3, or xyz = 0.
From this, I understand that the level surface will consist of the coordinate axes, but is there any way to parametrize or otherwise explicitly define this? If not, should xyz = 0 be sufficient as an equation of the level curve? Thanks!
 
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Well, from what I can tell, the equation xyz=0 seems to be what their looking for.
 
It doesn't just consist of the coordinate axes (they don't even go through your point), it consists of three planes. Which plane goes through your point?
 

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