What does the [x/2] notation mean, here?
It's floor(_) - as in, floor(1.999) = 0, but floor(2.001) = 2. If you look carefully the upper flange of the [] square brackets is missing, which makes it a floor.
I liked this simple calculus exercise
41–50 of 163 posts
Re: I liked this simple calculus exercise
#42What does the [x/2] notation mean, here?
It's the floor function https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
(Had to check the TeX code to figure that out, MathJax let you do that with a right click on the equation.)
Re: I liked this simple calculus exercise
#43Great post! It really drives home the point that understanding the core concepts in calculus is way more important than just memorising formulas and mechanically applying them. The example problem shows how visualising and breaking down a seemingly complex integral can actually reveal its simpler underlying structure.
This reminds me of the need to be adaptable and versatile when tackling math problems, since relying solely on known techniques can limit your ability to solve more complex or unfamiliar problems.
Educators should help students focus on developing a deep understanding of math concepts and honing problem-solving skills, rather than just bogging them down in calculations.
Re: I liked this simple calculus exercise
#44What does the [x/2] notation mean, here?
It's floor(_) - as in, floor(1.999) = 0, but floor(2.001) = 2. If you look carefully the upper flange of the [] square brackets is missing, which makes it a floor. https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
> floor(1.999) = 0
it's 1 not 0
Re: I liked this simple calculus exercise
#45> If some expression looks complicated, try graphing it and see if you get any insight into how it behaves.
This is not always a good idea. Some functions have complicated behavior that makes them either plain hard to draw (e.g. sin(1/x) near 0), or reach very high values but also be near 0, or be otherwise tricky.
Re: I liked this simple calculus exercise
#46Here's another exercise (resp. exam question) that tests understanding: given a sketch of a curve in a graph, roughly sketch the derivative (or integral). The number of otherwise good students who go "but I can't do the derivative without the formula?" suggests we need more questions like this.
Re: I liked this simple calculus exercise
#47Ha ha, sadly this can be transformed into a symbol manipulation answer as well. I know because this (stated slightly differently) is one of the questions in my 12th standard (senior year high-school equivalent) Mathematics I class. Here's someone writing it out on video on a tutoring site https://www.doubtnut.com/question-answer/int050exdx-where-x-... You have to spot the period, but x - floor(x) is called "fractiona…
That's just an algebraic application of the same idea. I don't think it's sad.
Re: I liked this simple calculus exercise
#48I did well in high school math. These days, when something involving algebra, trigonometry, geometry etc comes up I feel like I have a good understanding of it but my calculus seems weak to non-existent. I'm not sure if it's how I was taught, how I studied it or something else but calculus always seemed like a huge step change in difficulty. That said, I love how this article gives practical hints on how to replicate…
I find personally that my math ability is set to approximately 2-3 levels "below" the highest level math I completed, and I've seen hold for others. I have an applied math bachelors so I've taken analysis, dynamical systems, and other high level math classes, but I find that the stuff that I actually remember at a level to pass undergrad exams is up to linear algebra or maybe a little more advanced. Of course if I were to relearn it'd be much faster, but years of being a software engineer have caused me to forget all that stuff.
Re: I liked this simple calculus exercise
#49When I saw the equation referred to as (*), I had a flashback to those problem sets with *hard and **harder problems. ** problems often required some real out-of-the-box thinking. I wasn't always able to solve those, but it was so satisfying when I did (usually after an hour or two of struggle).
Re: I liked this simple calculus exercise
#50Current Calc 2 student here. I would be braindead approaching this problem honestly, I don't think I'd even know how to begin; I'm hoping that's normal. Why would the exponent be equal to x/2 - floor(x/2) be equal to x/2 on the interval [0, 2)? And how does the graph of x/2 - floor(x/2) imply anything about the behavior of e^(x/2 - floor(x/2))? I'm hoping I just haven't learned enough yet?
Man, thank you for writing floor(x/2) because I've been looking at the original post and didn't know what those silly brackets are supposed to mean.