CADGME 2014

Dynamic Algebra in EpsilonWriter

 

This document contains the demonstrations made during the communication at CADGME 2014.

Additional lines are in blue and red. They indicate settings and actions to perform.

The demos were made with EpsilonWriter 2.401

Later versions may have a different behavior due to evolution of the software.

 

If you have this document as an HTML page, as it contains EpsilonWriter data, you can copy the document (Ctrl+A then Ctrl+C) and paste in EpsilonWriter to get the document with editable formulas. You can then apply the suggested actions.

Otherwise, you can get the html page at http://www.epsilonwriter.com/TMF/

 

You can download EpsilonWriter or use the applet at http://epsilonwriter.com  

 

1. Equivalent drag and drop

 

Pedagogical dynamic algebra: only basic steps

 

With the tab "Calculation", choose "Dynamic Algebra: Pedagogical" and "Explain calculations on the line".

 

2x/3

Move 2 in front of the fraction: this is a possible basic movement

 

(2+x)/3

Move 2 in front of the fraction: this is not a possible basic movement

 

2x=3

Move 2 on 3 : this is a possible basic movement

 

=============================================  

 

Calculation dynamic algebra: more powerful

 

With the tab "Calculation", choose "Dynamic Algebra: Strong".

 

Solve 2(x+2)=3

Move 2 on x+2 then 4 on 3 then 2 on -1

 

Factorize (x-3)(3x+2)+x(9x+6)

Move 9 in front of x and choose "Factoring HCF". Move (3x+2) in front of (x-3) .  Move 3x on x  

 

Simplify 18x^4(x-2)²/(12x(x-2)³)

Move 18 on 12 , x^4 on x² and  (x-2)² on (x-2)³

  

Solve 2x+3y+5z=4 and 4x-y-2z=6 and 6x+2y-3z=1  

Write a 3 on the line after. Move this 3 on the = sign of the second equation, choose "Multiplication of both sides". Select the first equation and move it on the = sign of the second equation, choose "Addition side with side". Solve the second equation for z by moving 14x then the - sign. Select the second equation and move it on the = sign of the first equation, choose "Substitution". Do a Ctrl+Click on the = sign of the first equation.  

 

2. External drag and drop

 

Substitution

x=2a+1 and y=a-2    

Select the above simultaneous equalities and move them on the fraction bar below. Ctrl+Click on the fraction bar.

(x-y)/(x+2y)

 

=============================================  

3. Completing the square and a²-b²

 

2x²-6x+4  

Select the expression and choose "Completing the square". Select the first 2 and move it a little to the left to factor out 2 . Select the a²-b² form and choose "Factorization with a²-b² ". Ctrl+Click on the adequate - and + signs to add the numerical fractions.  

 

4. Solving schemas

 

Solve x³+3x+2=0    

Select the equation and choose "Solve equation". You get the Cardan's formulas solving schema. Move the p= and q= on the = sign of Δ=  and choose "Substitution". Use Ctrl+Click to calculate Δ.  As Δ>0 , click on "ok" for the case Δ>0 and on the delete buttons for the two other cases. Select Δ=8 and move it on the = sign of x_1= , choose "Substitution". Ctrl+Click on the inner radicals to transform  √8 into  2√2 . Select q=2 and  move it on the = sign of x_1= . Ctrl+Click on the two fraction bar.

  

5. Suggestions

 

Solve x³+3x²-x+2=0  

Select the equation and choose "Solve equation". Select x=y-1 and move on the = sign of the equation below,  choose "Substitution". Ctrl+Click on the = sign of the obtained equation to expand and simplify. You are now in the situation 4.  

 

6. Tables of signs and solutions of an inequality

 

x(x-2)/(x+1)>0  

Select the inequality and choose "Table of signs filled and solution".

Select again the inequality and choose "Table of signs to fill".  

 

7. Definition conditions of a function

 

f(x)=1/(x+1)-2/(x-2)

Ctrl+Click on the = sign to get the definition condition then Ctrl+Click on and to get a definition set.

  

8. Limits  

a= limit _x→+∞ (3x³+4x-6)/(x²-1)

Ctrl+Click on lim  to get the application of a theorem of "highest degrees". Move x³ on x² to get a simplification.  Ctrl+Click on lim  to get the result.

 

b= limit _x→1^- (3x³+4x-6)/(x²-1)  

Ctrl+Click on lim  to apply the substitution x=1^-  Select the numerator and choose "Numerical calculation".  Select the denominator and choose "Numerical calculation". You get 1/0 which is  an indefinite expression. Delete this step by Ctrl+Z then select the denominator and choose "Calculation of a limit". You get 1/0^- .  Ctrl+Click on the fraction bar, you get the result.

 

9. Derivatives

 

f'(x)=(xsin(x)/(x²+1))^prime0

Ctrl+Click on the prime on the right of the = sign. Repeat this action until there is no prime.

 

10. Variation tables

 

f(x)=(-x-4)/((x-2)(x+1))    

Duplicate the formula (use the "Duplicate" button or copy and paste). Put a prime on f. Select the fraction and type prime.  

Ctrl+Click on the prime on the right of the = sign. Repeat this action on the prime obtained.  Select the numerator and expand and simplify. Select the exponent 2 of the denominator and move it through one parenthesis on the right. Select the numerator and apply "Completing the square". Select the numerator and apply "Factorization with a²-b² ". You get f'(x)=(x+4-3√2)(x+4+3√2)/((x-2)²(x+1)²) Ctrl+Click on the  = sign, you get the "Variation table". Remains to fill the limits and two values of the function.

  

11. Several presentations of the explanations

 

-2x>3

Select -2 and move it over 3.

With the tab "Calculation", choose "Explain calculations in a table".

Go back to the given inequality, select -2 and move it over 3.

With the tab "Calculation", choose "Explain calculations in a table".

Select -2 and move it over 3. There is now a mathematical explanation and a gesture description.