Geogebra
GeoGebra is an interactive package for learning and teaching mathematics and science. The name comes from Geometry + algebra, but the software supports statistics
and calculus as well. It is highly interactive and does not require (or
support) programming. This makes Geogebra very easy to use, but at the
same time, somewhat limited. Since it is so easy to use, Geogebra is
suited both for elementary and fairly advanced studies. The software is
open source and is written in Java. It is available for a large number
of platforms. Here is
the home page.
Solve the following problems using Geogebra.
To start the program, type: /chalmers/sw/unsup64/geogebra-5.0.18.0/geogebra-portable
This is quite a new version so it contains debug printouts, e.g.
the program would like to store info. in the install directory, but is
not allowed to do so. If you do not want to see the printout, redirect the output to /dev/null , so in tcsh
/chalmers/sw/unsup64/geogebra-5.0.18.0/geogebra-portable >& /dev/null
and in bash
/chalmers/sw/unsup64/geogebra-5.0.18.0/geogebra-portable &> /dev/null
The program is still a bit buggy, so do not spend
an inordinate amount of time searching for possibly nonexistent bugs in
your programs.
Answer to one common question: to see small numbers (all decimals), open the Options-menu, choose Rounding and make your choice.
Problem 1, elementary functions and sliders
Create a slider, a, in the interval (-1, 1). Plot the function exp(a x).
Right-click on the slider and switch on animation. Switch off animation
and right-click on the curve, and switch on Trace, move the slider.
Suppose you are teaching a calculus course and want to teach your
students some properties of elementary functions. Use the slider-tool
to create three sliders, one for w and one for a in the interval (0,
5), one for d in the interval (-10, 10). Create the functions f(x) =
cos(w x) and g(x) = a sin(w x + d). Form the sum of f and g and call it
s. Visualize the three functions, in the same plot. Use colour in a
good way.
Create a slider for n. Plot sin(x) and use the TaylorPolynomial-command to plot the Taylor-polynomial (around x = 0) of degree n = 1, 3, ..., 11 in the same plot.
Choose 3D Graphics from the View-menu. Try plotting some surfaces from your course in multivariable calculus.
Problem 2, some commands working on sets of points
Create a point-set by first clicking on the point button, and then
place the points by clicking in the draw area. Collect the points in a
set using the syntax, variable = {A, B, ...}, where A, B, ... are the names of the points. Test (at least) the following commands on your set: ConvexHull, DelaunayTriangulation and TravelingSalesman, Try moving the points.
Use the commands FitLine and FitPoly to fit polynomials of degree at most one and two to your set of points. Try moving the points.
Problem 3, An epidemic model for influenza (or perhaps ebola)
The population is divided into three groups (and corresponding
functions wrt to time): those susceptible of catching the disease,
S(t), those infected, I(t), and those that who are recovered and
immune, R(t). In case of deadly diseases one would have to add a fourth group, deceased, D(t).
A common and simple model is the SIR-model (from S, I, R). In this
model we assume that the influenza has a short duration (weeks and not
years), so births and deaths are ignored. There is no latency period,
so once a person has been infected he or she can infect another person.
A recovered person is immune and cannot infect another person. Here is
the system of ODEs:
S' = -b S I, I' = b S I - c I, R' = c I
where b and c are constants. We need initial values as well. For
details about this system, read (part of) the Wikipedia article: Compartmental models in epidemiology.
Visualize S(t), I(t) and R(t) using Geogebra. You should use sliders
for b, c, S(0) and I(0). Be prepared to show the solution for b =
0.0025, c = 0.4, S(0) = 500, I(0) = 1 and the time interval [0, 25]
(since I know what the solution should look like in this case :-) Use
colour and text in a good way. You can use, the customary, beta and
gamma instead of b and c instead (Geogebra supports Greek letters). Use the NSolveODE-command for solving the system.
