This applet illustrates the main features of the normal or gaussian distribution.
Showing posts with label Mathematics. Show all posts
Showing posts with label Mathematics. Show all posts
Wednesday, April 3, 2013
Monday, December 24, 2012
Riemann sum
This applet illustrates the concept of Riemann sum, and shows how it converges to the Riemann integral when the number of partitions grows.
The Riemann sum, or numerical integral is shown in green, and the exact integral is shown in red.
Please notice that GeoGebra is not very powerful at integrating, so, for some functions, the exact integral will crash.
The Riemann sum, or numerical integral is shown in green, and the exact integral is shown in red.
Please notice that GeoGebra is not very powerful at integrating, so, for some functions, the exact integral will crash.
Friday, November 23, 2012
Vector field
This applet allows easy visualization of a two dimensional vector field. The equations for both components can be edited. The length of the plotted vectors can also be changed in order to visualize them more neatly.
Vector addition
This applet demonstrates the addition of vectors using the parallelogram rule. The vectors can be edited by dragging the blue dots.
Sunday, October 28, 2012
Complex multiplication
This applets illustrates graphically the concept of complex multiplication. As one can easily see, the multiplication of two complex numbers can be performed graphically by multiplying both modules and adding the phase angles.
Saturday, October 27, 2012
Sampling and aliasing
This applet illustrates, in a very simple way, the concept of aliasing. If the sampling frequency is inadequate, the reconstructed signal will be completely erroneous, as can be seen, for example, in the initial setup.
Friday, September 14, 2012
Bézier curves
This applet traces Bézier curves of different orders. The number of control points can be edited, and each control point can be moved by drag and drop.
Thursday, May 24, 2012
First order linear differential equation
This applet plots the slopefield of any linear differential equation, and tries to integrate and plot the solution (given the initial condition).
GeoGebra is not very good at integrating, so it will fail in displaying the exact solution very frequently. The slopefield can always be displayed, even for complicated equations.
GeoGebra is not very good at integrating, so it will fail in displaying the exact solution very frequently. The slopefield can always be displayed, even for complicated equations.
Wednesday, May 9, 2012
Number e
This applet demonstrates how the number e can be found by imposing equality between an exponential function and its derivative.
For wich value of a are the two functions identical?
For wich value of a are the two functions identical?
Tuesday, April 3, 2012
Integration
This applet illustrates the concept of definite integral of a function, as the area (with sign) under f(x) and between the vertical limits x = xi and x = xf.
Related applets:
Differentiation.
Related applets:
Differentiation.
Monday, March 26, 2012
Differentiation
This applet illustrates the concept of first derivative, as the best of a series of succesive approximations to the slope in a point.
Friday, March 23, 2012
Poisson distribution
This applet represents some features about the Poisson distribution, very useful to describe the probability of events wich occur with a known average rate and independtly of the occurrence of the last event, like nuclear decay processes or even car crashes.
Sunday, March 18, 2012
Polar coordinates
This applet plots a function specified in polar coordinates, being x the angle in radians.
Wednesday, March 14, 2012
Eigenvalues and eigenvectors
This applet is based in the one on Linear mappings, being essentially the same with the option of displaying eigenvectors and eigenvalues.
The black vector represents the input of the linear application, it can be edited by click and drag. The red vector represents the result of the mapping. The mapping matrix can also be edited. Remember that not all linear mappings have real eigenvalues.
Questions:
· What happens if the black vector lies over an eigendirection? Is there any relation with the eigenvalues?
· Introduce the following matrix a = 0, b = 1, c = -1, d = 0. Move the vector and try to guess what the transformation represents. It has no real eigenvectors, why?
· Is it possible for a two dimensional linear mapping to have more than two eigenvectors? Why?
· What happens if we transform the application to a basis of eigenvectors?
The black vector represents the input of the linear application, it can be edited by click and drag. The red vector represents the result of the mapping. The mapping matrix can also be edited. Remember that not all linear mappings have real eigenvalues.
Questions:
· What happens if the black vector lies over an eigendirection? Is there any relation with the eigenvalues?
· Introduce the following matrix a = 0, b = 1, c = -1, d = 0. Move the vector and try to guess what the transformation represents. It has no real eigenvectors, why?
· Is it possible for a two dimensional linear mapping to have more than two eigenvectors? Why?
· What happens if we transform the application to a basis of eigenvectors?
Monday, March 12, 2012
Coordinates
This applet aids to visualize the concept of coordinate and the concept of change of basis in linear algebra.
The red vector is editable by click and drag, and so are the basis vectors.
Observe that not only the orientation of the new base vectors is important, but also the module.
Try to insert two linearly dependent vectors as generators of the base B. You'll get an error message. Why?
The red vector is editable by click and drag, and so are the basis vectors.
Observe that not only the orientation of the new base vectors is important, but also the module.
Try to insert two linearly dependent vectors as generators of the base B. You'll get an error message. Why?
Taylor series
This applet shows the first 5 Taylor polynomials of a given function. The function is editable, and also the center of the neighborhood.
Observe how the successive orders converge to the given functions.
What happens if the function is a polynomial of order 5 or less?
What happens if the function has discontinuities, like tan(x)?
What happens if the function is not differentiable, like abs(x)?
Observe how the successive orders converge to the given functions.
What happens if the function is a polynomial of order 5 or less?
What happens if the function has discontinuities, like tan(x)?
What happens if the function is not differentiable, like abs(x)?
Sunday, March 11, 2012
Binomial distribution
The binomial distribution describes the probability of obtaining some number of successes in a sequence of n independent yes/no experiments. For example, the probability of obtaining exactly 3 heads in 5 rounds of coin tossing is P(5;3,0.5), being 0.5 the probability of success in a single round.
Saturday, March 10, 2012
Linear mappings
This applet shows the behaviour of a linear mapping in two dimensions. The image vector is shown in red, and the black vector can be displaced by click and drag.
The transformation matrix is editable.
Questions:
1. Construct the image of the vector (0,0). What happens?
2. Construct the image of the vector (1,1), and afther that, the image of the vector (2,2). What happens? Why?
3. Can you estimate, just playing with the vector, some eigenvector? Remember that not all linear mappings have real eigenvectors.
Related applets:
Eigenvalues and eigenvectors.
The transformation matrix is editable.
Questions:
1. Construct the image of the vector (0,0). What happens?
2. Construct the image of the vector (1,1), and afther that, the image of the vector (2,2). What happens? Why?
3. Can you estimate, just playing with the vector, some eigenvector? Remember that not all linear mappings have real eigenvectors.
Related applets:
Eigenvalues and eigenvectors.
Sines and cosines
This applet allows you to visualize the sine and cosine of an angle inside a goniometric circumference.
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