Discussion Overview
The discussion revolves around the function f(x,y) = x² - 2y² and its contour curves. Participants explore whether these curves are hyperbolas or hyperbolic paraboloids, and also delve into finding the tangent plane at a specific point on the surface defined by the function.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Homework-related
Main Points Raised
- One participant asks about the contour curves of the function and whether they are hyperbolas.
- Another participant suggests setting x² - y² = k and drawing the curves in the xy-plane to analyze them.
- Some participants express uncertainty about whether the curves are hyperbolas or hyperbolic paraboloids.
- A later reply asserts that the surface is a hyperbolic paraboloid and that the contour curves are hyperbolas.
- Multiple participants inquire about finding the tangent plane at the point (√2, 1, 0) and discuss the derivatives involved, particularly df/dz.
- One participant points out the ambiguity in considering df/dz since the function does not explicitly include z.
- Another participant suggests redefining the problem in terms of g(x,y,z) = f(x,y) - z to clarify the derivatives.
Areas of Agreement / Disagreement
Participants express differing views on the nature of the contour curves, with some asserting they are hyperbolas while others consider the possibility of hyperbolic paraboloids. The discussion regarding the tangent plane also reveals uncertainty about the appropriate derivatives to use.
Contextual Notes
There are unresolved questions regarding the definitions and assumptions related to the derivatives, particularly df/dz, and the nature of the curves derived from the function.
To find the tangent plane at the point (sqrt(2),1,0),I have to find df/dx,df/dy,df/dz...Is df/dz=d(x^2-2*y62-z)/dz=-1?Or df/dz=dz/dz=1?