Need Help with a Tricky Homework Problem? Find Solutions Here!

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Homework Help Overview

The discussion revolves around a problem involving the rotation of a function, specifically z = e^(-x^2), about the z-axis and its implications for the resulting equation in three-dimensional space. Participants are exploring the mathematical representation of this rotation and its geometric interpretation.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning how to express the rotation of the function as an equation and what it means geometrically. There are attempts to clarify the relationship between the original function and the resulting equation after rotation, with some participants suggesting that each point on the curve generates a circle.

Discussion Status

The discussion is ongoing, with participants providing guidance on how to articulate the problem and explore the implications of rotating the function. There is a focus on formalizing the mathematical expressions involved, but no consensus has been reached on the exact formulation or justification of the approach.

Contextual Notes

Participants have noted a potential misunderstanding regarding the initial problem statement and have pointed out the need for clarity in the definitions and assumptions related to the rotation of the function.

diffusion
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Homework Statement


problem.jpg


Not sure how to approach this one, does anyone have an idea?

BTW, in the top line it should read xz-plane, not xy.
 
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Hi diffusion! :smile:

(try using the X2 tag just above the Reply box :wink:)

Forget all the stuff about the cylinder …

the question is simply to prove that if you rotate z = e-x2 about the z-axis, you get z = e-x2-y2 …

that's easy, isn't it? :wink:
 
tiny-tim said:
Hi diffusion! :smile:

(try using the X2 tag just above the Reply box :wink:)

Forget all the stuff about the cylinder …

the question is simply to prove that if you rotate z = e-x2 about the z-axis, you get z = e-x2-y2 …

that's easy, isn't it? :wink:


I'm sure it's easy once I've been set in the right direction. I can see that z = e-(x2-y2) by graphing it, I just don't know how to show mathematically that it does.
 
ok … what does "rotate z = f(x) about the z-axis" mean? … how would you express the instruction as an equation? :smile:
 
tiny-tim said:
ok … what does "rotate z = f(x) about the z-axis" mean? … how would you express the instruction as an equation? :smile:

Well you will rotate the function around the z-axis, so that z = f(x) becomes z = f(x,y). Not sure how to express it as an equation, besides telling you that the equation is a function of both x and y.
 
Try putting it in words first …

what happens to each individual point on the original curve?​
 
tiny-tim said:
Try putting it in words first …

what happens to each individual point on the original curve?​

Produces a circle when rotated.
 
diffusion said:
Produces a circle when rotated.

Yup! And the equation of a circle is … ? :smile:
 
tiny-tim said:
Yup! And the equation of a circle is … ? :smile:

Yep, I recognized this before I posted the question. Again I just don't know how to show it, or justify it. I suppose I would say something like

"Since each point on the graph z = e-r2 will generate a circle when rotated about the z-axis, and the equation for a circle is r2 = x2 + y2, we can make this substitution into our equation for r2, giving us z = e-(x2+y2)."

I don't know, somehow this justification seems vague and inadequate.
 
  • #10
diffusion said:
"Since each point on the graph z = e-r2 will generate a circle when rotated about the z-axis, and the equation for a circle is r2 = x2 + y2, we can make this substitution into our equation for r2, giving us z = e-(x2+y2)."

ok, let's formalise that …

the point (x0,0,z0) generates the circle (x,y,z), where … ? :smile:
 
  • #11
tiny-tim said:
ok, let's formalise that …

the point (x0,0,z0) generates the circle (x,y,z), where … ? :smile:

At z0? Sorry, not sure if I really understand what you're asking. Are you asking for a point, the equation of the particular circle generated, or something else?
 
  • #12
I'm asking for the x,y,z equation(s) of the circle generated by the initial point (x0,0,z0). :smile:
 

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