mubashirmansoor said:
Lets imagine that we are talking about extremely nearby points where the three inserted terms belong to three neighboring points, the tangent to the curve at the middle point should be such that the angle between the curve and the tangent is equal for both neighboring points.
That's just not true. The concept of a "neighboring point" doesn't make much sense on its own either.
When you add ANY two things together and divide the result by two you are averaging them. (A+B)/2 is the average of A and B.
mubashirmansoor said:
We know that differentiation is itself not accurate and that we say “delta x” approaches zero which is impossible, and the real answers should be a little bit more than that of differentiation.
I have no idea what you mean by that. When you differentiate something, you take the limit. It is not an approximation.
mubashirmansoor said:
I thought to decrease the size of the curve (not the dimensions) by 10^10, in this way 1 unit would turn to 10 billionth of a unit and even the points with huge gaps between each other would become extremely near.
Even if the tangent function behaved linearly under multiplication of its argument, this would still just give you the
same result. However, the tangent function does
not behave linearly under multiplication of its argument, so doing this just changes your result so that sometimes it will be better and sometimes it will be worse. Try using your expression on, say, e^x. The one with those 10^10 factors will give a worse result. I'd actually be very surprised if it always gave a better result for polynomials too.
Sometimes your technique will give you a better answer than just averaging the slopes of secants. But not always (and probably not even
usually), and in order to get that precision you need to be able to compute tangents and arctangents to high precision
as well as finding the slopes of the secants to high precision. Adding and dividing by two is
much easier, if you need a quick approximation!
mubashirmansoor said:
I don’t feel that the technique has a problem but its answers are absolutely the same as differentiation for degree 2 equations
They aren't absolutely the same. It's still an approximation, you just aren't calculating it to enough places to see the difference. For example, calculating
10^{12}\tan{\left[\frac{1}{2}\left[\mbox{arctan}\left(\frac{2.01^2-4}{10^{10}} \right)+ \mbox{arctan}\left(\frac{4-1.99^2}{10^{10}}\right)\right]\right]
gives a result of 3.9999999999999999999999999996000000000000000000001 (to 50 places).
On the other hand, just averaging the slopes of secants (taken over the same range in x on both sides) will
always give the
exact derivative for any second-order polynomial.
(and note that in order to do that calculation without a computer, I'd need a table of arctangents of arguments on the order of 10^{-12}!)