That technique won't scale if you're trying to integrate some crazy complex function.
Integrating a piecewise continuous function by splitting it is used all the time.
I liked this simple calculus exercise
71–80 of 163 posts
Re: I liked this simple calculus exercise
#72Here'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.
Also a good question to test intuition is being asked to sketch e^sin(x) and sin(e^x).
Seems a little much for a regular calculus class. Wouldn't that fit better in a complex analysis class?
Re: I liked this simple calculus exercise
#73Current 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".
Also called "fractional part", which makes the symmetry a bit more simply obvious.
The tricky part of the problem isn't calculus, it's in a bit of algebra that often isn't emphasized in school.
Re: I liked this simple calculus exercise
#74As with so many integration exercises, this seems more like an exercise not in integrating, but in your understanding of the relevant functions.
Re: I liked this simple calculus exercise
#75Earlier quoted context omitted.
I was literally about to write this same comment. I did physics so I finished some group theory and complex analysis, but 20 years later all that stuff is gone because I never applied it in other classes. Only the stuff that kept coming up, like calc 1/2 and first order diffeq, really stuck.
With that said, does anyone know of a good method to relearn math efficiently? I found it to be really hard to self-learn any math topic, most books repeat everything from the basics at the beginning like what a set is, and then suddenly turn into ultra-advanced with “the proof is trivial” all around.
I think this experience is typical of self teaching from a math textbook. It's extremely difficult to find a good book that leaves no gaps while simultaneously explaining everything that might be difficult to understand. The key thing when encountering this for me is to expand my horizons and begin looking for videos or other supplementary materials. A teacher would show you the proof, or at least help you along the way, the internet can be used for this as well, up to a point.
While it's frustrating and time consuming, self teaching a difficult subject is just like that unless you're a god amongst men (and few of us are). Sometimes you'll want to fight through the strange unproven thing by thinking hard for a couple weeks about it while googling intermittently to find key steps. Sometimes you'll have to give up on it and keep moving. If it's foundational then fighting through can be highly beneficial, but a lot more things are presented as foundational than actually are.
I'd recommend finding good books by searching relentlessly on reddit and other forums for opinions, dedicating the time necessary to self teach something difficult (it can take upwards of a year to get through a smaller textbook if you have other things in your life going on), and if you really want it then fight for it. Give it everything you have, really let the problem consume your thoughts because eventually you'll wake up at 3am and know exactly what to do. And finally, move on if you don't want to do that. Try just keeping moving. Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts. Or you could find something you want to know and work backwards through every term that's used until you're at a concept and then attempt to apply it to the larger idea.
In general, self teaching math is extremely difficult, and only really works if you're willing to dedicate the time to fight through ideas.
Re: I liked this simple calculus exercise
#76Slightly off-topic: didn't know what ⌊x/2⌋ is Google: x squared (???) GPT: The expression ⌊x/2⌋ represents the greatest integer that is less than or equal to x/2. It is called the floor function of x/2. For example, if x=5, then ⌊x/2⌋ = ⌊5/2⌋ = 2. If x is an even integer, then ⌊x/2⌋ = x/2. If x is an odd integer, then ⌊x/2⌋ = (x-1)/2.
I was aware of the floor function (and the corresponding ceiling function) since I’m a software engineer. But I wasn’t aware that you could graph it. It never came up in high school or college math. And I never thought about it. Of course, it makes sense now that I’ve seen it.
It's also sometimes referred to as a step-function though i believe that name encompasses many more functions than just the floor
Re: I liked this simple calculus exercise
#77Current 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 think this kind of problem is less about maths and more about how one might approach an unfamiliar problem. Not knowing where to begin is normal. What you're looking to do is to build up the intuition for how you can break down the problem into smaller pieces so that you can investigate its properties. x/2 - floor(x/2) is the natural place to start because it's the smallest independent piece of the equation. Take a…
You don't need graphs or area at all, of course. All you need is to notice that the frac(x/2) expression is a periodic function that is only integrable piecewise, and is easy to integrate with a change of variables for each piece or a handwave of such.
It's actually gnarly to write out a formal proof as a new student would do (it requires principle of induction to handle all the pieces), but easy for an expert to breeze through as trivial.
The makes it a bit of an unfair problem for a students trying to follow the rules of math. This is very common challenge for students making the transition to higher math, when they are taught rigorous proofs but before they learn that professionals mathematicians are rarely rigorous (except when there is disagreement about the truth of an "obvious" claim).
Re: I liked this simple calculus exercise
#78Slightly off-topic: didn't know what ⌊x/2⌋ is Google: x squared (???) GPT: The expression ⌊x/2⌋ represents the greatest integer that is less than or equal to x/2. It is called the floor function of x/2. For example, if x=5, then ⌊x/2⌋ = ⌊5/2⌋ = 2. If x is an even integer, then ⌊x/2⌋ = x/2. If x is an odd integer, then ⌊x/2⌋ = (x-1)/2.
I was aware of the floor function (and the corresponding ceiling function) since I’m a software engineer. But I wasn’t aware that you could graph it. It never came up in high school or college math. And I never thought about it. Of course, it makes sense now that I’ve seen it.
Can you name a single-variable function that software engineers use that can't be graphed?
How would such a function be useful?
Re: I liked this simple calculus exercise
#79Earlier quoted context omitted.
I was aware of the floor function (and the corresponding ceiling function) since I’m a software engineer. But I wasn’t aware that you could graph it. It never came up in high school or college math. And I never thought about it. Of course, it makes sense now that I’ve seen it.
It's also sometimes referred to as a step-function though i believe that name encompasses many more functions than just the floor
It's piecewise-linear and periodic. Also called sawtooth. Step functions are piecewise constant. It would not be fun to climb steps of this shape.
Re: I liked this simple calculus exercise
#80Earlier quoted context omitted.
It's the floor operator. Also known as round down.
And introduced by Iverson for APL in 1962: https://www.johndcook.com/blog/2021/04/15/floor-ceiling-brac... and https://en.wikipedia.org/wiki/Floor_and_ceiling_functions .
APL, famous for baffling programmers with weird notation, also known for making one part of math more readable :-)