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Old-School Slashdotter Discovers and Solves Longstanding Flaw In Basic Calculus (mindmatters.ai)
Longtime Slashdot reader johnnyb (Jonathan Bartlett) shares the findings of a new study he, along with co-author Asatur Zh. Khurshudyan, published this week in the journal DCDIS-A: Recently a longstanding flaw in elementary calculus was found and corrected. The "second derivative" has a notation that has confused many students. It turns out that part of the confusion is because the notation is wrong. Note -- I am the subject of the article. Mind Matters provides the technical details: "[T]he second derivative of y with respect to x has traditionally had the notation 'd2 y/dx 2.' While this notation is expressed as a fraction, the problem is that it doesn't actually work as a fraction. The problem is well-known but it has been generally assumed that there is no way to express the second derivative in fraction form. It has been thought that differentials (the fundamental 'dy' and 'dx' that calculus works with) were not actual values and therefore they aren't actually in ratio with each other. Because of these underlying assumptions, the fact that you could not treat the second derivative as a fraction was not thought to be an anomaly. However, it turns out that, with minor modifications to the notation, the terms of the second derivative (and higher derivatives) can indeed be manipulated as an algebraic fraction. The revised notation for the second derivative is '(d 2 y/dx 2) - (dy/dx)(d 2 x/dx 2).'"
The report adds that while mathematicians haven't been getting wrong answers, "correcting the notation enables mathematicians to work with fewer special-case formulas and also to develop a more intuitive understanding of the nature of differentials."
The report adds that while mathematicians haven't been getting wrong answers, "correcting the notation enables mathematicians to work with fewer special-case formulas and also to develop a more intuitive understanding of the nature of differentials."
I appreciate the new form is technically more accurate but the expansion is pretty large compared to the original form... I wonder if the extra length doesn't wash out the understandability gains you get out of the original form.
"There is more worth loving than we have strength to love." - Brian Jay Stanley
dy/dx doesn't represent instantaneous rate of change. That would be nonsense. The d in dx and dy means "small difference that will eventually go to zero". This is why dy/dx is a fraction. It represents the limit of a small change in y divided by small change in x, as the changes go to zero. This is why we teach students about the limit definition of the derivative as being what the derivative really is. As far dy and dx being tricks of notation, they're really not. They really are small changes. There's no instantaneous rate of change. dy and dx are always finite real numbers. They never actually become zero. dy/dx is the ratio that is approached as they get smaller and smaller.
As far as this guy's new version of the second derivative, I call bullshit. I seriously doubt that this is correct. And the notation d^2y/dx^2 actually makes sense when you think about. It's really just d(dy/dx)/dx, that is, a small change in dy/dx divided by a small change in x, where dy/dx is a small change in y divided by a small change in x. Writing it in the other way is just a good way of doing it. If you draw out what this means graphically, is becomes clear that it's really a small change between two consecutive small changes in y divided by two small changes in x, that is d(dy)/dx^2, hence d^2y/dx^2.
This guy's new version, on the other hand, doesn't make sense at all. I mean, how do you get that from taking the derivative of the first derivative. Let's take a pretty standard function: x=1/2*t^2+2*t+12. x'=t+2; x''=1, whereas his version would be x''=1-t, which doesn't make any sense, unless he has completely redefined everything. I mean, d^2y/dx^2 would have to be something like 2t+5 and d^2x/dx^2 would have to be something like 2, and then we get x''=2t+5-(t+2)*2. I didn't read the paper so I don't know what it would actually be, but there's no doubt that x''=1, so if his method is to make any sense at all it would have to give the same results in the end. I just don't see how it could.