The other day I had a fairly popular post talking about how mathematicians easily and often admit that they don't know things or don't understand things. Today at work a real-life example came up!
Original post linked to in the next toot since apparently I can't post a link and have an image at the same time ... wtf?!
I was helping a student with Calc I in my office. The question gave a function and asked for values of x where the tangent line was horizontal. The function is the first in the image.
This requires taking the derivative with the product rule. The result of this is the second in the image. Since the second term has a denominator (other than 1 of course) we need to combine the two terms so we can set the numerator to 0 and solve.
The result of this operation is the third in the image. Fractions are 0 when their numerators are 0, so the fourth line shows the equation to be solved.
The student got this far without any help but was unable to solve the equation. This is commonplace. After all, the hardest part of calculus is algebra. But I couldn't see how to solve it either, so I told the student this.
At this moment two of my colleagues were talking in the hall outside my office so I told the student I'd ask them about it. Neither knew how to solve it and told me as much. So I told the student, who was actually thrilled that none of us could solve it either.
So I asked Wolfram Alpha, which gave a solution using the Lambert W, aka the productlog function. I'm a combinatorial topologist -- I do graph theory of various kinds. I've heard of this function but otherwise know nothing at all about it. And I'm happy to admit it! Anyway, that's how mathematicians roll.
ETA: Of course this problem shouldn't have appeared in an introductory calculus text since no undergraduate at that level would be able to solve it, so its inclusion was a mistake of the author or the editor.
#Calculus #LambertW #HorizontalTangent #DifferentialCalculus #Math #Mathematics #Mathematicians
Alright, future engineers!
A **Limit** is the value a function *approaches* as its input gets infinitesimally close to a specific point. Ex: `lim(x->0) sin(x)/x = 1`. Pro-Tip: Evaluate from both sides to confirm the limit exists!
"This first odor differential equation does not pass the smell test."
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics—and they are mostly clever fools—seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
—Calculus Made Easy, by Silvanus Thompson
It's great to see such a huge turnout for this past weekend's No Kings Rally. However, at the end of the #weekend, a friend (who works in healthcare) asked me which rally I went to. I told him I didn't attend any due to caregiving responsibilities. He asked what was so difficult about bringing our son to a rally.
So...I want to talk about #caregiving for a bit. I've mentioned here before my son is fully #disabled with severe spastic #quadriplegic cerebral palsy, is #actuallyautistic, and has significant #sensory #needs. Caregiving is physically and emotionally #brutal. It's a significant #life change that, unfortunately, a lot of #people simple do not and cannot fathom - unless it happens to them.
The best way I can describe *my* #experience is it's real time #multivariable #calculus with known, unknown, and constantly changing variables. It requires high level non-linear systems-based thinking to solve #problems, #chaos #management, and constantly being on high alert.
#Burnout is inevitable, the #physical and #emotional #labor is genuinely exhausting on a cellular level, and #time and #access to self care is rare. It's also incredibly #socially #isolating.
In addition to the #social #isolation, navigating the outside world isn't just unnecessarily difficult it's a #failure of the #system. The uncomfortable #reality is most places are NOT #accessible, including #bathrooms. #Toileting is a basic #human #right! My son is too large and heavy for the baby changing stations in "#family #restrooms." We need to put him on a #dirty bathroom floor in order to change his #diaper. Where is the #humanity and #dignity in *that*?
I feel like a broken record explaining this to him for the past six years. It's honestly mind blowing and exhausting.
Nuevo video en mi canal, ahora es sobre integrales de línea: https://youtu.be/Q2a2ccmzz3Q
#physics #física #matemáticas #mathematics, #calculus #cálculo #integrales #integrals
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