I liked this simple calculus exercise • Hacker News Board

Re: I liked this simple calculus exercise

#21

Post by jrumbut » Mon, Apr 17, 2023, 6:55 AM UTC

nateb2022 wrote: post #6

Current 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?

It's definitely a tricky problem for a student. The reason that everyone likes it is that this is exactly the kind of problem you run into when you need to solve an integral in the real world.

Once a year at least I run into a math situation like this. Obviously in some professions it will be much more (or less) often.

Exponents, floor, ceiling, and absolute value are very frequently part of the problem.

The approach of graphing the function, breaking it into components, and seeing if any of them are periodic, are all important steps toward a solution (more so than the symbolic manipulation because that might either be a big mess or even unavailable).

Often you'll end up using numerical methods to approximate the solution, but if you can come up with a closed form solution that's much nicer.

Re: I liked this simple calculus exercise

#22

Post by ajeet_nathawat » Mon, Apr 17, 2023, 6:56 AM UTC

I loved maths in school and unfortunately didnt pursue maths but solved this problem while sipping coffee and listening to cornfield chase, I realised why I loved maths. the solution is so simple and so intuitive if you solve with graph.

Re: I liked this simple calculus exercise

#23

Post by aashutoshrathi » Mon, Apr 17, 2023, 6:58 AM UTC

We used to get lot of such tricky stuff during the preparation of IIT-JEE here in India, and I'm telling you if you don't understand Area under curve is integral, you can't touch most of the questions. But I get your point, if you are interested in such questions, you should checkout IIT JEE mathematics question, you'll love them

Re: I liked this simple calculus exercise

#24

Post by Scarblac » Mon, Apr 17, 2023, 7:01 AM UTC

nateb2022 wrote: post #6

Current 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?

I see the expression "x/2 - floor(x/2)" and automatically read it as "the part of x/2 to the right of the decimal point".

Re: I liked this simple calculus exercise

#25

Post by lordnacho » Mon, Apr 17, 2023, 7:06 AM UTC

navels wrote: post #15

Reminds me of college when I said to my Real Analysis professor "that's a neat trick". His response: "It's not a trick, it's a method." :-)

I would classify a trick as something that happens to work but isn't rigorous. Like treating dy/dx as a fraction sometimes works, but only under certain conditions.

Re: I liked this simple calculus exercise

#26

Post by lordnacho » Mon, Apr 17, 2023, 7:08 AM UTC

This is why I enjoyed doing math contests. You always got these problems that illuminated how things actually work, and the answer is always some set of basics applied to elegantly solve it.

I'm actually looking for a set of such problems as I think it's a lot better than grinding out hundreds of quadratics or polynomial derivatives and such. I found the AOSP stuff already, wonder if there's other good sources.

Re: I liked this simple calculus exercise

#27

Post by de6u99er » Mon, Apr 17, 2023, 7:18 AM UTC

Nowadays I use Wolfram Alpha for stuff like this.

Re: I liked this simple calculus exercise

#28

Post by xivzgrev » Mon, Apr 17, 2023, 7:18 AM UTC

laurieg wrote: post #12

I 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 think it’s a common feeling. Even though they are both “math”, they feel like different skill sets.

Re: I liked this simple calculus exercise

#29

Post by cperciva » Mon, Apr 17, 2023, 7:19 AM UTC

First rule of integrating non-analytic functions: If they're analytic everywhere in the interval in question except a finite number of points, split the integral and compute it one analytic segment at a time!

(Second rule: If the function is non-analytic at an infinite number of points, you probably still want to compute it one segment at a time, but adding them back together afterwards may get messy.)

Re: I liked this simple calculus exercise

#30

Post by actionfromafar » Mon, Apr 17, 2023, 7:22 AM UTC

nateb2022 wrote: post #6

Current 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?

I was guilty of the other extreme - I had a very hard time understanding the symbol pushing, so I tried to find numerical tricks all the time, which didn't work out too often.