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

In summary, the problem is to find an equation of the level surface that passes through the point (1,-2,0) for the function f(x,y,z)=xyz+3. The constant for the level surface is found to be k = xyz + 3 = (1)(-2)(0) + 3 = 3 and the resulting equation is xyz = 0. This equation consists of three planes and the question remains which of these planes goes through the given point.
  • #1
adartsesirhc
56
0
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!
 
Physics news on Phys.org
  • #2
Well, from what I can tell, the equation xyz=0 seems to be what their looking for.
 
  • #3
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?
 

1. What is a "level surface" in Calc 3?

A level surface in Calc 3 is a three-dimensional graph that represents a function of two variables, where the output of the function is constant. In other words, all points on the surface have the same value for the function.

2. How do you find a level surface at a specific point?

To find a level surface at a specific point, you first need to set up an equation with two variables that represents the function. Then, you can plug in the coordinates of the point into the equation and solve for the constant value. This will give you the equation of the level surface at that point.

3. What is the process for solving a Calc 3 problem involving finding a level surface?

The process for solving a Calc 3 problem involving finding a level surface includes setting up the equation for the function, finding the partial derivatives with respect to each variable, setting the derivatives equal to zero, solving for the constant value, and finally, plugging the constant value back into the equation to get the final level surface equation.

4. What are some common mistakes when solving a Calc 3 problem for a level surface?

Some common mistakes when solving a Calc 3 problem for a level surface include not setting the partial derivatives equal to zero, making calculation errors, and not considering the domain of the function when solving for the constant value.

5. How can I check if my answer for a level surface problem in Calc 3 is correct?

You can check if your answer for a level surface problem in Calc 3 is correct by plugging the equation of the level surface into the original function and checking if it yields the constant value at the given point. Additionally, you can graph the equation on a three-dimensional graph and visually confirm that it is a flat surface at the given point.

Similar threads

Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
470
Finding the velocity of flow described by a vector field
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Find all points where surface normal is perpendicular to plane
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Multivariable calculus problem
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Finding the volume surrounded by a curve using polar coordinate
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
How should I solve this Diophantine equation word problem?
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Perturbation Methods: Asymptotic Matching Question
    • Calculus and Beyond Homework Help
Replies
2
Views
713
Proving that a curve intersects a surface at a right angle
    • Calculus and Beyond Homework Help
Replies
8
Views
6K
Unit Normal to a level surface
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Finding functions that are a level set and a graph
    • Calculus and Beyond Homework Help
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top