many integration problems can be done by a method
called "derivative divides" which can be taught to
students, but is not.
It removes any heuristics or pattern matching from
about 90% of calculus problems. Using neural networks
and such seems to me to be a really bad way of approaching
these problems.
Since the patterns are not valid grammatically, you will
have to do some work if you wish to use them. For example,
"d/dx" clearly does not mean 1/x by cancelling the d's. :)
RJF
Jurgis Pralgauskis wrote:
> thanks for Your answers,
> as much as I found out, Maple has exactly what I was looking for
> http://adept.maplesoft.com/categories/maple9/html/Calculus1.html
>
> And Mathematica proposes some pattern-matching
> http://documents.wolfram.com/mathematica/book/section-2.3.14
>
> well but I thought for a while, and I think, it wouldn't be too hard
> to implement formula pattern matching according to ~10 main symbolic
> integration rules
> and parse the problems according to them.
> http://www.alcyone.com/max/reference/maths/integrals.html
> matching would actualy be done by solving comparatively_simple
> equations (so it would not eat CPU - I hope;)
>
> first ratexpand should be run, to go through trivial rational
> simplifications
> and then integration rules' matching should begin
> some optional sequence of aplying rules would help the solving :)
> there would be a tree of tries to match,
> and once a way on the tree is found, it is shown step by step to user..
> expert systems (prolog, CLIPS) usually work this way, as I know..
>
> I thought through an example of integrating (x/(x*x+1), x)
> and I don't see any big problems on this way (CPU usage bothers mostly)
>
> ---
> another strategy would be orientied to deal with divisor , if f(x) =
> a(x)/b(x)
> because such cases take most of time, tracking , which formula to apply ;)
> so factor (b(x) +1) and factor((b(x) - 1) should be checked to see
> if there is a posibility of applying arctan, arcsec or arcsin
> because other way it is hard to spot such possibilities..
>
> neural network could help to choose the strategies, as it can be learned
> overtime...
>
> hm, well, just a small question who has time to make this :)
> me - no earlier than summer... so for now - just a small dream
>
> good luck everyone..
>
>> hi,
>>
>>> I generaly know, what Maxima is up to,
>>> but I am interested is it possible to track the solver as the problem
>>> is beeing solved
>>> can I get step by step output (as we are taught in calculus studies)
>>> instead of just the final answer?
>>
>>
>>
>>
>> I don't know if there is anything built in Maxima that would do this
>> (other than trace command that Professor Fateman mentioned), but for
>> simple, or not too complicated, that I sometimes used Maxima in class
>> for, I would insert some steps myself. Say, we want to find a formula
>> for the inverse of sinh(x).
>> (%i1) y=(exp(x)-exp(-x))/2;
>> (%i2) 2*%;
>> (%i3) subst(z,exp(x),%);
>> (%i4) z*%;
>> This one would not be expanded, which I prefer; I can ask students
>> to complete the next step, and check it with Maxima, ...
>> At my current institution, we are using Mathematica, but I did not
>> see a simple way to do above with Mathematica; in fact several
>> students asked me the very first lab class if it would be possible to
>> force Mathematica to show more steps. Some more experienced
>> Mathematica colleagues showed me a couple of ways, but those never
>> looked as simple and intuitive to me as the above stuff with Maxima, so I
>> already forgot them:)
>> Milan
>>
>