The purpose of this activity is to develop the procedure for the analysis of a function, usually done with pen and paper, using GeoGebra for computations and for the representation of data.
Students are free from computational burden, and reflect on the meaning: the meaning of each step of the procedure, on the sequence of steps, on the interpretation of the results as they are obtained, and on the final synthesis.
Clearly, the graph of the function should not be made visible to the student until the activity has ended and to check if the solution is correct. This can be done by defining the function as auxiliary object not shown. Thus, the function does not appear in the algebra window and the graph is not represented. The function to be studied can be displayed as LaTex text in the graphics window.
Basically, we only require four commands, a function and a GeoGebra tool: the Root and Intersection commands and the function Sgn (to determine the domain, to find the zeros and to study the sign) and the commands Limit and Derivative and finally the instrument Pen.
Protocol
No. Name ==> Definition / Command
1 Function f
2 Text 1
3 Text 2
4 Function g ==> g(x) = sgn(f(x))
5 Point A ==> (0, f(0))
6 Point B ==> Root of Numerator[f]
6 Point B2 ==> Root of Numerator[f]
7 Point R ==> Root of Denominator[f]
8 Number LimDx ==> LimitAbove[f, x(R)]
9 Number LimSx ==> LimitBelow[f, x(R)]
10 Line a ==> x = x(R)
11 PenStroke ==>
12 Number LimInf ==> Limit[f, ∞]
13 Number m ==> Limit[f(x) / x, ∞]
14 Number q ==> Limit[f(x) - m x, ∞]
15 Line b ==> y = m x + q
16 PenStroke
17 Function f' ==> f'(x) = f'(x)
18 Point D ==> Root[numerator[f’(x)]]
18 Point E ==> Root[numerator[f’(x)]]
19 Function h ==> h(x) = sgn(f'(x))
20 Point F ==> (x(D), f(x(D)))
21 Point G ==> (x(E), f(x(E)))
22 Function f'' ==> f''(x) = f''(x)
23 Function p ==> p(x) = sgn(f''(x))
24 PenStroke
25 Function r ==> f
