The re-usable part, that you've called a "script", can be implemented in Maple as a user-defined procedure.

Afleiding_Euler_knik_2_staven_inc_rotveerRev_3_ingekort_mapleprimes_ac.mw

You may use the numapprox package to construct a rational polynomial that approximates your f (up until its slope gets acceptably small...).

We can start from a small positive value, since not everything here behaves so nicely at x=0 exactly.

I did not raise Digits here. If you plan on exporting the resulting expression into some external application that can handle float double-precision (eg. compiled code) then you could try setting Digits:=15 and see if better accuracy attains (depending on degrees requested).

Download approx_ac.mw

Above I requested chebpade construct a polynomial (degree in denominator as zero). Often one can do better with a rational polynomial. There are also other commands in the numapprox package which might be useful to you, depending upon your example and target criteria.

Download approx_ac2.mw

[edit] I gave this answer because of the following preceding detail from the OP. Up until this time there had been no mention by the OP of obtaining good asymptotic matching. As described below the posed problem had been about getting an accurate match up until some cutoff value:

"Let's say I still wanted to obtain an approximation for f(Gamma) for Gamma from 0 to that Gamma sending f(Gamma) relatively close to sqrt(5)/5...Such approximate polynomial should be a very simple (no more than one line) function of Gamma that well captures its original shape up to the point it starts stabilizing at sqrt(5)/5 (very roughly at which Gamma?). How would you do this?"

The goal of obtaining matching asymptotic behavior (as x tended to infinity), which can involve quite different methodology, was only agreed with later by the OP.

You have not shown us any definition of (or assignment to) xA||i and yA||i for i=1..6. You should explain that.

Download LA_Jamet_ex.mw

The cost of the overhead of wrapping an expression sequence in square brackets -- obtaining a list -- is small.

And then you can use either numelems or nops.

Download numelems_seq_list.mw

When none of your piecewise conditions holds it will evaluate to its default value, which by default is zero. That collides with your goal of asking implicitplot to find where it is zero.

You can pass an additional argument to the piecewise command, as some alternative default value.

Download pw_def_impl.mw

Is this the kind of effect you are after?

You could also adjust the granularity factor (below, 50). You could also make it match or multiple what you could pass for the frames option of the animate command (not done below).

You can remove the caption, naturally.

Download anim_ex2.mw

If you want the rotation to take some other route then please describe it explicitly.

Your methodology of trying to assign numeric values to delta as well as delta(m) is flawed.

Try changing the solitary delta instances to Delta.  Or make all the delta(m) into say deltam.

This is what your commands perform, using assignments instead of % for the referencing.

Note that your second use of % is referring to what is assigned below to T. Your second use of % is not referencing the same as your first use of % (which is the result assigned to S below).

Download indets_q.mw

What are you actually trying to query, via indets? Subexpressions (or instead perhaps only things of some specific type, such as names only?) in the solve result?

[edit, upon consideration that you might be asking about another aspect.]

Or perhaps you're asking about what counts as an indeterminate/subexpression? The result of indets(f(_Z),_Z) includes plain name _Z, the argument of f. But the common dummy _Z of RootOf is not returned, as a specially treated consideration.

Download indets_qZ.mw

The dummy _Z of RootOf is special in other ways. The dummy index name of sum/Int/etc is not changed or specially "uniquified", but RootOf turns such into _Z.

Download RootOf_note.mw

In a way it boils down to what developers have deemed useful. A query of whether an expression contains any RootOf ( eg, has,hastype or indets(...,RootOf) ) will be a test for the presence of its dummy name _Z. In contrast a query about plain name _Z used in some other context, eg. ln(_Z) alone, doesn't benefit from that implication.

But since the dummy name in sum/int/etc is user-defined and arbitrary then it make a bit more sense to report on it.

Download indets_q3.mw

You've set up the define_external call so that you can pass in matrix `c` as one of the arguments (and thus act on that inplace). So no matrix actually gets returned. This is reflected in the Fortran code shown.

So you might look in the documentation for how to specify a NULL return (or something ignorable, possibly integer[4]).

Here is one simple way to get interleaved text and 2D Math for output (without the quotes of the strings, and with the strings in upright roman rather than italices, etc).

This site uses a very old maplenet which renders the units in double-braces. In the actual modern Maple GUI those units get rendered as usual.

Download some_TS.mw

Try that instead as,

textplot([[coordinates(A)[], "A"], [coordinates(B)[], "B"], [coordinates(C)[], "C"]],
         align = {above, right});

Try right-clicking inside the Table, and then selecting Table->Properties in the popup. (Or open the GUI's right context-panel, and left-click to place the mouse focus in the Table.) Then change the "Page break" property to, say, "Allow between rows".

Then try exporting of print-to-PDF.

(If you actually have a Table with nested Tables inside each cell, say, then you need to be careful to position the mouse-pointer some place inside the outermost Table but not within any inner Table, so that you actually adjust the properties of the Table that has all the rows...)

The programmatic Table constructor of the DocumentTools:-Layout package allows pagebreak to be specified as an option. But that option is not offered by the Tabulate command, which utilizes that lower-level constructor. I'll submit a software-change-request that either the default be changed in Tabulate or that it offers the option itself.

An example of a programmatic way to get such an effect, without the need to right-click:

M:=Matrix(77,2,(i,j)->i*j):

xml:=subsindets(DocumentTools:-Tabulate(M,'output'=':-XML'),
                "pagebreak"="none",()->"pagebreak"="row"):
str:=DocumentTools:-ContentToString(
       DocumentTools:-Layout:-Worksheet(
         DocumentTools:-Layout:-Group(xml))):

## If you want the table alone in a new GUI tab
#Worksheet:-Display(str);

## If you want it embedded in this same worksheet
DocumentTools:-InsertContent(str):

The above code also works for me (page-breaking in PDF export/print) if M is a DataFrame instead of a Matrix.

You could use the Explore command for this.

For example (and without going all-out for optimizing it),

Explore_anim_ex.mw

Your followup example can be done as follows:

Download Jamet_1.mw

Yes, you can construct a reusable procedure from your expression involving the functions in the dsolve solution.

I cannot tell whether you want epsilon and omega*t (as you've written in your Question text) or varepsilon and `ωt` as you've used in the parameters list in your worksheet. I guess that it's the latter, but I've set it up below so that you can assign what you want to expr.

Download New_2_ac.mw