@acer Thanks for spotting that, and I've corrected it.

@Scot Gould I've lost sight of the main point of debate here. I thought that you were arguing that Maple had an "inconsistency" that led to a pedagogical flaw. Yet the example that you showed shows Maple working perfectly (in a pedagogical sense also). And, you can do that in both the top-down and bottom-up modes. If you were indeed arguing that there was a pedagogical flaw, then I want you to give an example that highlights that flaw.

You wrote:

• [C]ould not one devise a system, for which space is allocated, but all evaluation is performed numerically at the last step?

I guess you mean essentially this: Is it possible to write a symbolic system (such as Maple) in a non-symbolic language (such as C, C++, etc.)? Yes, of course it's possible since Maple itself is an example of that. But I think that your conception of what's involved is overly simplistic. If the variables don't have numeric values, you have to internally handle all the algebraic manipulations. You need to put expressions in "normal forms" so that equality can be recognized.

• If understand the statement in Maple, can you provide an example of a language they are referring to?

Almost all languages in widespread use before 1986 (such as C, C++, Fortran, Pascal, BASIC, Algol, COBOL) have no built-in support for symbolic variables. In those languages, the bottom-up evaluation style must be used.

You specifically mentioned Matlab and Python. As far as I'm aware, both of those languages have some support for symbolic variables. In Matlab, they're call "symms" (I believe). In Python, there's a package sympy. 

@J F Ogilvie The first of your three examples can be done like this:

inttrans:-fourier(
    convert(exp(-a*abs(x))/sqrt(abs(x)), Heaviside), 
    x, s
) assuming a>0;

For the other two examples, I can't get Maple to fully expand them, even by direct integration. But neither are the results incorrect, as was suggested by your original Question. It may be possible to include these in Maple's "known" fourier transforms by using the inttrans:-addtable command.

You may know that the expressions inside the csgns are real, but do you know wheher they are positive or negative?

How can you expect anyone to answer that when you've given no formulas, no code, and no reference material?

@Scot Gould Why do you refer to 2*x + 7 = x^2 + 4*x + 4 as the original equation? The original equation is sqrt(2*x+7) - x - 2 = 0.

@janhardo You can't divide the interval by n-1 if n=1. Acer's `if` is there to treat n=1 as a special case.

@mmcdara The second solution that I gave in my Answer doesn't require DataFrame, will work in Maple 17, and places the row labels on the right. So, there'll be no plot needed.

@Joe Riel According to Stefan's writing above

• [The bound variable] is quietly implicitly declared local if one wasn't declared

To me, that adequately explains why the protected problem no longer occurs. But it doesn't explain the absence of the assertlevel problem.

@janhardo If e1 and e2 are expressions and c is a condition that evaluates to true or false, then

if c then e1 else e2 fi

is often abbreviated

`if`(c, e1, e2)

Transposition is an involution (f(f(x))=x); it's not idempotent (f(f(x)) = f(x)). A linear idempotent function is called a projection.

@janhardo Yes.

Let me know if you need any mathematical hints to compute the x-interval needed to obtain a particular length of tangent line.

Why do you want the extrema of the spline rather than the extrema of the original data?

@janhardo I suggest making the x-ranges of the tangent lines (tangency point) +- (b-a)/2/n where n is the number of tangency points. Currently, you have n hard-coded as 6. If you want to improve this procedure, my next suggested step is to make this 6 a variable.

@janhardo Since a and b are now parameters of the procedure, they can't be locals. You simply need to remove them from the local list. While you're at it, you might as well add xaxis to the locals.

Then if you run the procedure, it'll work, but you'll see that the tangent lines are too long. Can you correct that?