Guessing plot of two-variable function

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SUMMARY

The discussion focuses on visualizing the plot of the two-variable function f(x, y) = √(x² + y²), which represents the distance from the origin to the point (x, y). Key characteristics include its positivity, symmetry regarding x and y, and its nature as an increasing function. The level curves of this function are circular, providing a clear geometric interpretation. Participants emphasize the importance of identifying symmetry, positivity, and behavior at infinity to aid in plotting such functions.

PREREQUISITES
  • Understanding of two-variable functions
  • Familiarity with metric spaces
  • Knowledge of level curves and their significance
  • Basic calculus concepts related to maxima and minima
NEXT STEPS
  • Explore the properties of level curves in multivariable calculus
  • Study the implications of symmetry in two-variable functions
  • Learn about the graphical representation of functions in metric spaces
  • Investigate the behavior of functions at infinity and its impact on plotting
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Students and educators in mathematics, particularly those studying calculus and metric spaces, as well as anyone interested in visualizing and analyzing two-variable functions.

twoflower
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Hi all,

recently we've been practising drawing a plot of two-variables functions (we're currently doing metric spaces).

Well, I don't know how to guess the plot for let's say:
[tex] f(x, y) = \sqrt{x^2 + y^2}[/tex]

One guy came to the blackboard and drew it quite quickly :frown:

Thank you for the suggestions.
 
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Some things you can see right away. The function is always positive. It depends only on [tex]x^2 + y^2[/tex] and is thus symmetric between x and y. The function is an increasing function of x and y. Basically this function is the distance from the origin to the point (x,y) and therefore its level curves are circles.

In general, look for symmetry, positivity, maxima, zeros, behavior at infinity, level curves, and anything you can easily identify (I'm not saying it is always easy to just see the level curves, max/min, etc but in this case it is).

Does that help?
 

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