Casyopee - Parameters (cursors)
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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

Graphing and Animating parameters

Expressing symbolic conditions at interval's bounds

Instanciating parameters

Substituting parameters

Three functions f, g and h are considered:

f is defined on (– ; – 1] by f(x) =  – 1 ;

g is defined on [ 1 ; + ) by g(x) =1 ;

h is defined on [ -2 ; 1] by h(x) = a x3 + b x ;

The goal is to find values of a and b connecting “smoothly” the three graphs and to create a piecewise defined function.

Create the functions f, g, h  (as in first example), entering the correct domains of definition. You will be prompted to accept the creation of parameters a and b.

Graphing and Animating parameters

Press the buttons  f, g, and h in the graphic tab. The graph of h is updated according to values of a and b

Initially a cursor is visible. You can make it (des)appear by clicking on the identifyer

                                  

Figure 50 -  cursors visible                                          Figure 51 - cursor invisible 

When the cursor is visible, you can animate the parameter with the mouse or the keyboard's arrows.

Figure 52 - Displaying functions f, g and h

By animating parameters, one can try to approach a solution. For instance with a=-1 and b= 2, the graphs are connected, but the connexion is not smooth. That is why we need to solve the problem symbolically.

Figure 53 - Graphs connected by parameter animation

Expressing symbolic conditions at interval's bounds

Verify that parameters are not instantiated  (see Figure 50 - Instantiated parameters in toolbar                                  Figure 51 - Formal parameter).

Press the button with x0 then x1 in the command bar at the bottom of the graphic tab.

 The symbolical value list helps to express algebraically the conditions.

 

Figure 54 - Value for x1 with parameters a and b

 

Figure 55 - Values for x2 with parameters a and b

Then the conditions are b +a = 1 (value in 1 and -1) and 3a + b = 0 (null derivative in 1 and -1);

By solving this system, you can obtain suitable values for a and b (a = -1/2 and b =3/2).

Note: Casyopée does not solve systems. Here a solution could be obtained by successively solving one equation in a, then substituting in the second equation and solving in b. Experts can use Evaluate Formula in the Algebra tab menu, and choose Maxima in Syntax for Entry.

Figure 56 - Solving a system with Maxima

You obtain a solution in the NotePad:  . Alternatively, you could have entered a formula like  solve([h(1)=1 , h’(1)=0],[a,b]).

Instanciating parameters in expressions

When calculating and resolving equations, Casyopée considers any parameter to be a symbolic value. This is why, by default, the parameters are not "instantiated" in the algebraic pane displays (list of expressions, values ​​of the variable,) and in the table of symbolic values. It is possible to instantiate one or more parameters in an expression thanks to the contextual menu (right click). When you check the parameter (s), the expression is displayed in a special way.

The values ​​or limits of a function displayed in the list of symbolic values ​​are instantiated according to the choices made for the expression defining this function.

Substituting parameters

Animate the parameters to give them the values of the solution and check on the graph. You may have to adapt the step, that is to say the increment for each move of the cursor; use the Parameters button, then in the Step box put 1/2 for the increment.

Select the function h, click on Calculate, choose substitute; parameters a and b values are taken into account.

Figure 57 - Menu Calculate / substitute

Figure 58 - Confirmation box for substitution

A new function h0, independent of the function h, has been created.

Figure 59 - Graphic displaying of substitution


Creation date : 05/10/2014 - 11h09
Category : - help
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