GeoGebra geometry and algebra software
June 10, 2010: In 2001 Markus Hohenwarter at the university of Salzburg started the GeoGebra geometry software project, intended for free use in schools.It has since evolved with assistance of what must be a very resourcefull community of programmers, translators, coordinators, moderators etc..In 2009 it won the prestigious Tech Award for benefiting humanity.
In addition to sketching classic geometry GeoGebra also will find slopes of, and areas under curves plus derivatives of many polynomial functions, and indeed - that is great stuff - for free!!.
Without much trouble GeoGebra will yield the integral (ie. area) of the difference between a cubic polynomial function f(x) and its derivative f´(x) in the interval defined by the two intersections of the plots of the functions.
As a layman I do not yet understand the deeper meaning of this area, but I might have seen something similar when I years ago went to school.
Using GeoGebra 3.2.41:
1. At the Input line enter f(x)=2x^3 + 7x^2 + 3x - 3
(the graph of the function will pop up.You can adjust grid, colour etc. as you wish under options and by right clicking the graph)
2. Enter derivative[f] or f'(x)
(the parabolic graph of the derivative pops up)
3. Then find the “Intersect Two Objects” tool in one of the drop down lists, and click on the intersections of the two graphs.GeoGebra now construct the intersect points A and B.
4. To calculate the area between the two curves enter integral[f,f',x(A),x(B)]
The area between the curves found this way is the definite integral of the difference f(x) - f'(x).
In the GeoGebra forum a relatively limited algebraic capability of GeoGebra is talked about.As I understand GeoGebra will not calculate square roots amongst other things.But very exitingly I also understand a beta version of the software incorporating a Computer Algebra System (CAS) is on its way.If true, this probably will make GeoGebra on par with the most advanced handheld graphic calculators you can buy, and again, for free!.
Tips: It is possible to edit the functions on the sidebar after double clicking, and graphs will then update automatically.The keyboard up and down arrows will print the history of previously entered commands on the input line.Be sure to use the correct apostrophe character in formulas, or things will not work.
To download this nice and attractive program search, and you will find.