Visualization of a numerical method for determining the type of local extrema of functions with two variables on a contour map without using derivatives
Further refinement of the applet for exploring contour maps. The presented applet visualizes both the contour map created by moving point B and the corresponding three-dimensional graph. 8 functions are provided as examples.
Representation of the function f(x,y) in a graphical 3D window. The applet contains a 3D image of the function f(x,y). The applet contains a 3D image of the function f(x,y). The "false" mode provides a more detailed image of the function in a small neighborhood of the test point N on the contour map, shifted in 3D graphics at the origin. As a test circle we have a circle of radius r with the center at the point N. kr -multiplicity of this radius. These parameters are used to adjust the x-y plane boundaries, the scaling parameter is used to adjust the function values.
Search for critical points. Many objects of the applet are associated with the point N and therefore it is difficult to move it on the contour map. For this purpose, it's "assistant" is provided - point A, which can be easily moved on the map to the desired locations. In this case, the point associated with it moves along the 3D surface f(x,y). Using the button :
N can be moved there. In this case, Test Point "Assistant" A returns to the origin O=(0,0) of the coordinates.
Using the available Zoom(+ A) Zoom(-A) buttons you can view the selected points of the contour map in more detail. By drawing the contour field each time, you can find the localization of critical points more precisely.
Determination of the critical point type. If there is a local maximum (minimum), the test point N is colored red (blue) and the test circle is colored blue (red): on this circle the function from the point N decreases (increases). If N is at the saddle point, it is colored green, and the test circle has red and blue areas of increasing and decreasing function, respectively.
*When switching applet modes, settings are often changed - “doubles” of circle segments often appear. In this case, click on the reset settings" button at the bottom of the left window and the settings will be restored. It is also appropriate to press the “2D zoom” and “3D zoom” buttons in case of failures.
**Explanatory pictures of this applet are given in the applet.
N can be moved there. In this case, Test Point "Assistant" A returns to the origin O=(0,0) of the coordinates.
Using the available Zoom(+ A) Zoom(-A) buttons you can view the selected points of the contour map in more detail. By drawing the contour field each time, you can find the localization of critical points more precisely.
Determination of the critical point type. If there is a local maximum (minimum), the test point N is colored red (blue) and the test circle is colored blue (red): on this circle the function from the point N decreases (increases). If N is at the saddle point, it is colored green, and the test circle has red and blue areas of increasing and decreasing function, respectively.
*When switching applet modes, settings are often changed - “doubles” of circle segments often appear. In this case, click on the reset settings" button at the bottom of the left window and the settings will be restored. It is also appropriate to press the “2D zoom” and “3D zoom” buttons in case of failures.
**Explanatory pictures of this applet are given in the applet. For a detailed discussion of Example 7 on the scheme for calculating stationary points of a function of two variables, see the link.
In the case of index=8, a new function is available for input. In this case, the "Initial Settings: Restore/Undo" flag must be set to "false" and the options must be selected manually: r, scaling, a, k_r.
I can recommend the following links from the Internet as exercises:
Find and classify the critical points
Functions of more variables: Local extrema
Extreme Values and Saddle
![[size=85]Displaying images of the function f(x,y) and its fragment.[/size]](https://www.geogebra.org/resource/bczpnatb/69hlC26Wc27kv67y/material-bczpnatb.png)