Vector-Valued Functions and GeoGebra

Zoran Trifunov, University "St. Clement of Ohrid", Bitola, Macedonia

Igor Dimovski,University for Information Science and Technology, Ohrid, Macedonia

International GeoGebra Conference for Southeast EuropeJanuary 15-16 2011, Novi Sad, Serbia

DEPARTMENT OF MATHEMATICS AND INFORMATICS

Aim & KeywordsThe aim of this talk is• to show some of possibilities of using GeoGebra in teaching

vector-valued functions in 2D; • to explain the needs of appropriate 3D approach.Keywords: • vector-valued functions; • parameterized 2D curve; • parameterized 3D curve; • derivatives; • tangent vector; • velocity vector;

GeoGebra

Sketching graph of real function y = f(x)Calculus courses at the universities we represent have included vector-valued functions and motion in plane and in space.

Using GeoGebra one can facilitate learning such concepts. Sketching the graph of a real valued function y = f(x) in GeoGebra is extremely easy.

Example 1a. Using GeoGebra to show the graph of the function

f(x) = x4 – 4x2.

- we can just type the previous equation into the input box, and the graph appears at once!

GeoGebra

Dynamic approach

Is it possible to show the graph on some other, more generic way?

In teaching mathematics, functional thinking is especially important. Very often there is a need of presentation of dynamic processes. Dynamic processes are necessary to explain with dynamic means, with presentations where movements – animations are shown.

Example 1b. Graph of the function

f(x) = x4 – 4x2.

can be parameterized using parameter a like this: a(a, a4 – 4a2)

GeoGebra

Advantages of that approach

Students are able to see connection between coordinates of the points of graph.

Students can see the motion of the point, and drawing up the curve, dynamically.

Students become able to feel the dependence of the variables, which is the most important during learning functions.

GeoGebra

Parameterized 2D curves

Learning parameterized curves and vector-valued function is important from several points of view.

Interpretation of vector function as a model of particle motion and its derivative as tangent vector of a curve that is a trajectory (path) that is described by that particle motion and the velocity vector of that motion, then the second derivative of that function as acceleration of that same motion etc., is very difficult to explain by the use of static graphics and almost impossible by means of the traditional teaching methods.

Here, dynamics become crucial. Definitely, animation is needed.

Parametrized 2D curves

For parameterized 2D curve, GeoGebra slider can be used to determine 2 coordinates of the point.

Setting the "trace on" option to the point, and also "animation on" option to the slider, one can obtain effective way for explaining the particle's motion in the plane.

It is not difficult to show the tangent vector to the curve, as effective way for visualization the meaning of the derivatives of the vector-valued function as a particle's velocity vector.

GeoGebra

Examples: Circle

Example 2.a

x(t) = 2cost

y(t) = 2sint, t [0, 2]

Those equations parameterize the circle.

Vector form:

r(t) = 2costi + 2sintj

The vector from origin to the particle’s position A(2cost,2sint) at time t is the particle’s position vector.

t OAr��������������

Examples: CircleExample 2.b

x(t) = 2costy(t) = 2sint, t [0, 2]

Derivatives:

x’(t) = 2sinty’(t) = 2cost.

The vector:

is the particle’s velocity vector, tangent to the curve.

At any time t, the direction of v is the direction of motion.

2sin 2cosd

t t tdt

r

v i j

Examples: CardioidExample 3. Particle’s position:

r(t) = (2cost – cos2t)i + (2sint – sin2t)j

x(t) = 2cost – cos2ty(t) = 2sint – sin2t t [0, 2]

Derivatives:

x’(t) = 2sint + 2sin2ty’(t) = 2cost – 2cos2t.

The vector:

is the particle’s velocity vector, tangent to the curve.

2sin 2cosd

t t tdt

r

v i j

Examples: ButterflyExample 4.

cos 5sin 2sin 4 sin12

t tx t e t

cos 5cos 2sin 4 sin

12t t

y t e t

[0,2 ]t

3D approach is needed

After all, it is clear why needed 3D approach for teaching vector valued functions is.

We tried to use GeoGebra 3D Beta to create instructional materials according to 3D parameterized curves, but we did not succeed. Maybe because we need powerful tool as, for example, Mathematica or Matlab software we usually use for that purpose.

Just to explain needed visualization power in 3D, we are going to show several animations created in Matlab.

Animations made in Matlab Matlab offers a quite easy (???) and interesting way of creating animations to help describe the dynamic features of functions.

The essence of creating such animations is in generating a great number of static pictures in which most objects remain unchanged, and only few changes. Every picture presents one frame.

A row of such frames is kept in the so called movie matrix that can be preserved as a separate file. Then, with a simple Matlab code an animation can be shown reading the film matrix, placing the static part of the graphics and issuing the command for showing the film. The created animation can be literary exported as a movie, i.e. video file in an .avi format.

Example

Example 5.

Example 6. Several video files produced with Matlab.

2 3

132

t tt t r i j k

Matlab

Movie 1

Movie 2

GeoGebra

Why GeoGebra 3D is needed?

… and to create a three dimensional Geometry and Graphics View in GeoGebra that is easy to use with the mouse.

This view will allow the creation and interactive manipulation of 3D geometrical objects like points, lines, polygons, spheres, and polyhedrons as well as function plots of the form f(x,y). The 3D View should both be usable in the GeoGebra standalone application as well as offer the possibility to be embedded into interactive web pages.

Because of its SIMPLICITY!

Vector-valued functions and GeoGebra

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