Level Surfaces Problem- Calc III

In summary: So, for any values of x, y, and z that satisfy z = 3x + 4y, the function f will equal 8. This will create a level surface for f = 8 that is the graph of the function 3x + 4y.
  • #1
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I have a couple of questions, but probably only need one worked out to figure out the rest.

1. Find a function f(x,y,z) whose level surface f=8 is the graph of the function 3x+4y

=> I know that a level surface for f(x,y,z) is the solution to f(x,y,z)=k. However, now I'm stuck. I know how two draw or identify level surfaces when they're given, but I don't know what this question is asking.

Solutions that I have tried that don't make sense:

3x+4y=8
3x+4y-z=8
A couple of other shots in the dark
...eh?


2. a)Given f(x,y)= sqrt(16-x^2), find level surface of g(x,y,z)=c representing f(x,y).
b) c=?

=> f(x,y) is obviously half a cylinder of r=4, but again I can't visualize what the question asks.


Thanks!
 
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  • #2
Andy13 said:
1. Find a function f(x,y,z) whose level surface f=8 is the graph of the function 3x+4y

Solutions that I have tried that don't make sense:

3x+4y=8
3x+4y-z=8

The first is an equation describing a subset of the plane; the second is an equation describing a subset of space. You need a function that assigns a value to every point in space. The only constraint on this function is that it should assign the value 8 to the points of the subset of space defined by the equation [tex]z = 3x + 4y[/tex], and not to any other points.

Here's a hint. If you know that [tex]z = 3x + 4y[/tex], what number can you predictably make out of [tex](x, y, z)[/tex]? How can you turn this number into the 8 you need?

Andy13 said:
2. a)Given f(x,y)= sqrt(16-x^2), find level surface of g(x,y,z)=c representing f(x,y).
b) c=?

Same strategy. You are asked to give a function [tex]g[/tex] and a number [tex]c[/tex] so that the [tex]c[/tex]-level set of [tex]g[/tex] is the subset of space described by the equation [tex]z = \sqrt{16 - x^2}[/tex]. Figure out a function [tex]g[/tex] which gives a predictable number at the points of this subset (and nowhere else).
 
  • #3
ystael said:
"The only constraint on this function is that it should assign the value 8 to the points of the subset of space defined by the equation [tex]z = 3x + 4y[/tex], and not to any other points.

Here's a hint. If you know that [tex]z = 3x + 4y[/tex], what number can you predictably make out of [tex](x, y, z)[/tex]? How can you turn this number into the 8 you need?"




Still having difficulty. What I get from your explanation of "constraints" is that 8 should be substituted for x and y; is this correct?

In which case, the number predictably gotten would be 56, which divided by 7 would give 8.

... right track? completely wrong? Thanks.
 
  • #4
Completely wrong, sorry.

Try a simpler problem. Can you give a function that takes the value zero in the [tex]xy[/tex]-plane (i.e., on the plane [tex]z = 0[/tex]), and nowhere else -- so that its 0-level set is that plane?
 
  • #5
... I would hazard a guess at no. Would it be permissible for you to numerically work though one of either my examples or yours, and we'll see if I can get the subsequent examples from looking at that that? Verbal explanations clearly aren't working (though I appreciate them!).

From your example: does that mean that, for z=0, the function also equals 0?
 
  • #6
ystael said:
...
The only constraint on this function is that it should assign the value 8 to the points of the subset of space defined by the equation [tex]z = 3x + 4y[/tex], and not to any other points.

Here's a hint. If you know that [tex]z = 3x + 4y[/tex], what number can you predictably make out of [tex](x, y, z)[/tex]? How can you turn this number into the 8 you need?
...

Andy,

Think about ystael's hint.

If [tex]z=3x-4y[/tex] then, what is [tex]3x-4y-z[/tex] equal to?
 
  • #7
To Sammy:

Oh, I misunderstood the previous wording.


3x-4y-z=0

So, to "turn this number into the 8 you need," add 8 to both sides?

hence f(x,y,z) = 3x-4y-z+8
 
  • #8
Andy13 said:
To Sammy:

Oh, I misunderstood the previous wording.


3x-4y-z=0

So, to "turn this number into the 8 you need," add 8 to both sides?

hence f(x,y,z) = 3x-4y-z+8

Yes, I believe that's what ystael was suggesting.
 

1. What is the Level Surfaces Problem in Calculus III?

The Level Surfaces Problem in Calculus III involves finding the equation of a surface in three-dimensional space that represents points with the same value of a given function.

2. Why is the Level Surfaces Problem important in Calculus III?

The Level Surfaces Problem is important in Calculus III because it allows us to visualize and analyze functions in three-dimensional space, which is crucial in many fields such as physics, engineering, and economics.

3. How is the Level Surfaces Problem solved?

The Level Surfaces Problem is typically solved by first setting the given function equal to a constant value and then using algebraic manipulation and techniques from multivariable calculus to find the equation of the surface.

4. What are some common techniques used to solve the Level Surfaces Problem?

Some common techniques used to solve the Level Surfaces Problem include partial derivatives, directional derivatives, and the gradient vector.

5. Can the Level Surfaces Problem be extended to higher dimensions?

Yes, the Level Surfaces Problem can be extended to higher dimensions, such as four-dimensional space, by using similar techniques and principles from multivariable calculus.

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