Level Surfaces Problem- Calc III

[Andy13]

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!

 

[ystael]

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?

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).

 

[Andy13]

"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.

 

[ystael]

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?

 

[Andy13]

... 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?

 

[SammyS]

...
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?

 

[Andy13]

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

 

[SammyS]

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.