Calculus Can Be Simple!

Uploaded by CalCuteLus on 04.08.2011

You probably heard of the new approach to calculus from 2010.
Well, I did, and I decided to google it..
So I stumbled upon this textbook and I read it cursorily.
I think it’s worth attention, and being inspired from Vihart’s informative math videos, I’d like to tell you what it’s about, shortly.
So the author presents there a different definition to the derivative.
Notice that if we take the expression f(x+h)-f(x) and let h go to 0,
it wouldn’t be such a great idea, since this limit is always zeroed for continuous functions.
Now, at the definition of the derivative this problem is resolved,
by first comparing the term f(x+h)-f(x) to h, via a fraction,
and only then applying the limit process.
So the author suggests that it is also possible to compare the term f(x+h)-f(x) to a constant,
in this case 0, via the “sign” function, prior to applying the limit process.
He suggests to name this pointwise operator “detachment”, and he gave it a weird semi-colon notation.
Note that the detachment is sort of a discrete derivative,
in the sense that the set of values it accepts is finite.
The geometrical interpretation of the detachment is kind of simple:
For example, at this point the detachment from right is +1, and the detachment from left is -1;
and at this point, both one-sided detachments are -1.
But wait a minute, you say – isn’t the detachment merely the sign of a function’s derivative?
(in that case, it’s not such an original idea after all..)
Well, that’s what I thought, however the answer is: not always.
Those examples show that the detachment is still capable of supplying information on the function’s monotony behavior,
in cases where the derivative is zeroed, non-existent, and even for discontinuous functions.
So the detachment IS indeed an original idea after all..
but what is it good for?
In terms of information level, true, the information regarding the tangent’s slope is lost at the definition of the detachment,
however as we saw, in some cases a tangent to the function’s graph does not exist or it may not be informative,
and we’re still capable of analyzing the function’s monotony behavior just by applying the detachment.
In terms of numerical approximation, the author claims that the detachment is numerically efficient, and it makes sense,
but he doesn’t verify it experimentally. Perhaps someone else should.
The detachment operator is not linear, however it does depict a multiplicative property.
And other properties: the detachment is not reversible (unlike the derivative, which can be reversed via applying integral);
and an extension of the detachment is defined in advanced theories of classical analysis, such as metric spaces, where the derivative is undefined.
So to conclude, I don’t know if the detachment should be part of the calculus,
but I think that it is a nice idea, and it should gain the proper attention.
I’d like to know what you think. Please leave a comment and let’s discuss it.
Take care, and while you’re out there, don’t drink and.. detach!