Table of variations
Study of a ln function using different software.
Used material : 
Derive4 / HP49Gv1.17-8b / TI-89v2.04
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Table of variations Study of a ln function using different software. |
Author : Olivier Miclo |
| Public : Students |
Used material : Derive4 / HP49Gv1.17-8b / TI-89v2.04 |
It does not seem necessary to me to show an example of resolution with a traditional function (of the form f(x)=a*x+b).
There are not indeed need for software or program for a such study.
The example which follows makes intervene the function log-Napierian, and shows the some difficulties met with 3 softwares : TI-89v2.04, HP49Gv1.17-8b, and Derive4.0.
The screens and the form of the present results on this page can change according to the version of the software.
f(x) = ln(( x^3+1)/x^3 )
How to carry out this study using a C.A.S. ?
First of all, let us ask a graphic representation to the TI-89. This will give us a global idea.
Unfortunately, this is only one visualization.
It is preferable to make a rapid study with the different commands proposed :
One can of course calculate without problem the derivative : see the results returned on TI-89 v2.03 and HP49G v1.17-8b :
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On TI-89, the result is simplified and adapted to the student. |
On HP49G, it will be necessary to ask a simplification of this expression.
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The first result returned by the HP49G is not that until one waits. This automatic not-simplification is not very pleasant in this example.
Simplification is a large problem during the conception of a C.A.S., whatever it is.
On HP49G, the designer preferred to return the results in "rough form", in order to leave the user free for the simplifications : he can thus triturate and apply such or such simplification to an expression as good seems to him. This can be dangerous for a high-school pupil who did not know the " logic " of this machine, but it is so much pleasant to be " the Master " of handling...
Study of the field of definition of the function ln( (x^3+1)/x^3 ) :
(x^3+1)/x^3 must be > 0, ...so ... x<-1 and x>0.
The derivative is calculable for all x element of R*: it is thus definite on [- 1,0[, contrary to the function ln((x^3+1)/x^3).
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On HP40-49G v1.18, a root is obtained for the derivative of our function, 0... whereas the TI-89 v2.04 returns an empty list, what is logical because the derivative is not definite for x=0 (see denominator)
The result returned by the HP49G is rather strange : the expression obtained via the function expand did not let us predict a such result. |
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| In fact, it is necessary to apply SOLVE to the "expanded" derivative and not to the rough derivative which can return additional solutions (SOLVE cancels the numerator of the expression without being concerned with the denominator) |  |
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Preceding calculations are confirmed with the program "DEFINIT" for TI-89 of Philippe Fortin. The none-simplified expression "(x^3+1)/x^3>0 " is due to the software v2.04 of the TI-89 not know to solve the inequations of order >2. |
And Derive ? (in its version 4)
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#1: definition of our function
#2-3: roots of the equation: no answer... it is logical, just like on TI-89 and HP49G.
#4-5: calculation of the derivative.
#6-7: roots of the derived one: two found solutions. The result returned by Derive resembles more of a limit calculation in + and - the infinite one which would lead to 0. |
The HP49G, v1.18 lays out of an integrated table of variations . On TI-89, we use the program " FX "
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TI-89 and HP49G return a correct table. On HP49G, read " - 1-0 " like " less 1 less ".
Let us note that on HP49G :
- the first line of the table corresponds to the various breaking values of X, with the sign of the derivative f'. Between -1(-) and 0(+), the table indicates an indetermination, whereas the derivative is defined in this interval.
- In 0(+), F takes the value " infinite " without signe (what is different from + / - infinity). This lets predict (to the user) an indetermination in this point. Go to HOME, type " 1/0 "... and see the significance of infinity without sign !
General remark: ln((x^3+1)/x^3) can be defined between -1 and 0, but with complex values, but for such a function, say that F is increasing or decreasing on this interval would have no sense.
Is it possible to check (and obtain) the sign as of the these expressions (F and f') with a C.A.S ?
Let us carry out this verification with a TI-89 v2.04 and a HP49G v1.17-8b
| Sign derivative |
What can be interpreted like " sign(x^4+x) ", positive for all x>0 is, and negative for all x<0. |
What can be interpreted as "sign(expr/(3*abs(expr)))", so "sign(expr)", what is equivalent to positive for all x>0, and negative for all x<0. |
| Sign ln((x^3+1)/x^3) |
The TI-89 refuses to evaluate this calculation, even by using the operator " knowing that ". |
What can be interpreted like "sign(expr/abs(expr))", that is to say "sign(expr)". (see higher for the field of definition) |
In this comparison, it is clear that neither the TI-89, nor the HP-49G gives " direct " result. It is necessary to interpret.
The FX algorithm & the HP49G make it possible to obtain the good table and to see the nondefinite zone in spite of the few difficulties encountered " with the hand " with the TI-89 and HP49G integrated functions.
The version 1.17-8b of the HP49G is an evolution compared to the preceding versions, because it detects now the fields of definition.
O-Miclo, 08 june 2000
