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Re: Re: What to do in v. 6 in place of Miscellaneous`RealOnly


On 26 May 2007, at 17:40, Helen Read wrote:

> David W.Cantrell wrote:
>> Helen Read <read at math.uvm.edu> wrote:
>>> Suppose my calculus students want to plot x^(1/3), for say  
>>> {x,-8,8}. The
>>> problem, of course, is that Mathematica returns complex roots for  
>>> x<0.
>>> In past versions of Mathematica, we could get the desired real roots
>>> (and plot the function) by loading the package  
>>> Miscellaneous`RealOnly. I
>>> guess we can still do it that way (and ignore the "obsolete package"
>>> message), but is there a suggested way of doing what we need in 6.0?
>>
>> Perhaps have them define their own
>>
>> realCubeRoot[x_]:= Sign[x] Abs[x]^(1/3)
>>
>> which plots as desired, of course.
>
> Well, yes, but it's kind of a pain to have to define their own root
> functions this way on an individual basis. (Not to mention, it
> completely hoses the derivative. Try realCubeRoot'[x] or
> realCubeRoot'[-8] and see what you get.)
>
> I was hoping for a more convenient way to do this in Mathematica 6.0.
> Surely it *knows* the real nth roots of x for n odd and x<0. It  
> would be
> nice to be able to define f[x_]=x^(1/3) or x^(3/5) or whatever and  
> just
> set some option to make it return the real value for x<0.
>
> --
> Helen Read
> University of Vermont
>


Yes, it knows it, and there is a way of expressing it, but I am sure  
you won't like it.

realCubeRoot[x_] := Root[#^3 - x &, 1]

Now,

realCubeRoot[-1]
-1

etc. Plotting is no problem. Moreover you can differentiate, but only  
using D !

  D[realCubeRoot[x], x] /. x -> 8
  1/12

It won't work with Derivative :

realCubeRoot'[x]
0

And just in case anyone thinks of reporting the last output as a bug  
to WRI, this has been known ever since RootObject appeared in version  
3 of Mathematica.

Andrzej Kozlowski






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