# Visualizing Functions of Two Variables. Version 3

- Author:
- Roman Chijner

*This Applet can be used to Visualizing Functions of Two Variables onto its xy-projection.*It in comparison to the previous version: https://www.geogebra.org/m/qjvkbXxP, is changed and the

**loading time has decreased.**The implicit method now have two possibilities of colouring Contour lines. Colouring is done without the lists.

__Here are explored__*:*

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**Extrema lines**

**.***Points of intersection of these lines are critical points.*

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**Two methods of drawing contour lines:****Implicit method**

**.**

*In this case colours of painting are dependent on the heights with respect to the x-y Plane. Lines of the same color are contour lines -similar to the*

*lines*

*of*

*equal thickness*

*in*

*Optical Interference*

*. You can use the "Frequency" parameter to adjust the distance between the lines. Increasing the frequency, you can examine in more details the functions region with small changes of coordinates z.*

**Explicit method**

**.**

*In this case contour lines are drawn directly on a x-y Plane. Both methods can be used for all points in x-y Plane and for a small area near the selected point.*

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**Critical points***of the function y=f(x,y) and its projection onto the x-y plane.*

*Attention: You can download the ap*

*plet to your computer and Java applet will run faster.*Images: https://www.geogebra.org/material/show/id/pxXTcawv