Casyopee - Model and locus from a parameter
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What they say:   I used Casyopée to help students solve optimisation problems. It allowed me to present students very open tasks. Students explored and found results that they had to prove afterward. Casyopée also helped students reuse basic strategies for solving.   A teacher
What they say:   Casyopée is faster and more convenient than a calculator.... We have the geometric and algebraic side of the problem at the same time. It is easier to see how a function "reacts." It's useful and interesting.   A student
What they say:   Casyopée makes it easy to calculate a derivative, to factor, to calculate zeros... and have a graph of the function next to it in the same window. It allows on a geometric problem to be able to establish variables that can then be used to study the problem by way of functions...   A student
What they say:   Casyopee is a powerful application that can prove useful to both students and teachers It allows you to use various exploration and modeling tools, with the purpose of studying or teaching mathematical functions.   Softpedia
What they say:   Casyopee comes with lots of features. One of these features is the help provided for proving a function. There is also a feature for writing HTML reports that include the mathematical functions. Casyopee is guaranteed to improve the mathematical knowledge of its users.   phpnuke.org
What they say:   Besides the concept of number, the concept of function is the most important one in mathematics      David Hilbert
What they say:   The notion of function is present in all scientific disciplines, and also in everyday life. Our experience as a teacher shows every day that it is a problem for many students. Situations with Casyopée can also be used outside of a technological environment and everyone will be able to reflect on her professional practice.   A university teacher

Suppose that some points depend on a parameter. It is interesting to obtain the locus of one point and to model dependencies between variables involving this parameter.
This is an example.

Consider a "Target" moving at uniform speed 2 on a straight line, and a "Chaser" moving at uniform speed 3, making an angle a in degrees relatively to the trajectory of the "Target ".

The two mobiles have an initial distance of 2, the "Chaser" being on a perpendicular to the trajectory of the "Target".

How to model these mobiles? For what values ​​of a, does the " Chaser " reach the "Target"?

We first create two parameters, t for time, and a for the angle in degrees, as shown in the adjacent picture. We create T, coordinated point (2t, 0) for the "Target" and C for the "Chaser" of coordinate : (3cos(ap/180) t ; 2-3sin(ap/180) t).

The trajectory of C is a straight line. This can be confirmed by using the menu entry "locus".

Please click on the tool in the menu (1 in adjacent figure) and then to (2), then the parameter (3).

Casyopée confirms that the locus is included in a line and proposes to create it.

 

This done, it is interesting to give different values ​​to a.

With a value of 30 °, the chaser passes before the target without hitting it. With a value of 60 °, it passes behind.

Casyopée’s automatic modeling will allow studying the variation of the distance between the two mobiles.

We create, c0 and c1, calculations corresponding to the time and distance.

The function f is obtained by automatic modeling. Its graphical representation appears strictly above the x-axis except for a value of a close to 48 °. In this case, it looks like the graph of a piecewise linear function and the distance is close to zero for a value of t close to 0.9.

Another consideration may be based on the observation that C is at a distance 3t from its initial position, that is to say on a circle of radius 3t centered on this position. By creating this circle, we can animate t so that it passes through point T. It is again a value of t close to 0.9 and an angle close to 48 °.

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By trigonometry we find latex: frac{2}{3} for the cosinus of a and latex: frac{sqrt{5}}{3}for the sinus. For these values, the formula of f is

latex:text{f}(t)= sqrt{5cdot t^2-frac{12cdot sqrt{5}}{3}cdot t+4} and simplifies into latex:text{f}(t)= |sqrt{5}cdot t-2)|

The above conjectures are proved.

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Creation date : 01/12/2014 - 17h06
Category : - help
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