I liked this simple calculus exercise • Hacker News Board

Re: I liked this simple calculus exercise

#11

Post by saghm » Mon, Apr 17, 2023, 5:11 AM UTC

ryan-duve wrote: post #2

This reminds me of an exercise I'll never forget from my Math Methods course: finding the derivative of arcsin(x). It seems almost impossible because, just looking at it, there seems to be nothing you can do to simplify it. Then, out of sheer nothing-else-to-do-ism, you take the sin() of it and realize sin(arcsin(x)) = x. Take the derivative of both sides, apply chain rule and draw a right triangle and you have the a…

In turn, this reminds me a bit of a calculus problem I saw freshman year of college that I ended up sharing with my old high school calculus teacher because it similarly looks intractable until you make a "simplification". The problem was to find the integral of `x/(x + 1)` (I forget the exact bounds, presumably from 0 to x with respect to x). The trick is that this the same as `(x + 1 - 1)/(x + 1)`, which you can then split into `(x + 1)/(x + 1) - 1/(x + 1)`, which you can then integrate much more easily.

Re: I liked this simple calculus exercise

#12

Post by laurieg » Mon, Apr 17, 2023, 5:11 AM UTC

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 the insight and solve the question, rather than just the insight itself.

Re: I liked this simple calculus exercise

#13

Post by mydogcanpurr » Mon, Apr 17, 2023, 5:23 AM UTC

ryan-duve wrote: post #2

This reminds me of an exercise I'll never forget from my Math Methods course: finding the derivative of arcsin(x). It seems almost impossible because, just looking at it, there seems to be nothing you can do to simplify it. Then, out of sheer nothing-else-to-do-ism, you take the sin() of it and realize sin(arcsin(x)) = x. Take the derivative of both sides, apply chain rule and draw a right triangle and you have the a…

The approach is less mysterious if you recognize it as going back to definitions, which is a common problem solving technique.

Re: I liked this simple calculus exercise

#14

Post by arjie » Mon, Apr 17, 2023, 5:28 AM UTC

Ha 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 "fractional part of x" where I come from and is a named function which everyone is familiar with. Then, without knowing the area-under-the-curve interpretation, one can blindly apply another symbol-manipulation tool: the summing of integral over a period.

Re: I liked this simple calculus exercise

#15

Post by navels » Mon, Apr 17, 2023, 5:56 AM UTC

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." :-)

Re: I liked this simple calculus exercise

#16

Post by mkl » Mon, Apr 17, 2023, 6:03 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?

Another way to think about it: floor rounds down, so there are intervals of numbers which all have the same floor, so you can break the problem up into intervals where floor(x/2) is constant. On [0, 2), floor(x/2) = 0, on [2, 4), floor(x/2) = 1, on [4, 6), floor(x/2) = 2, etc. So in each interval you are integrating e^(x/2) multiplied by e^0 or e^-1 or e^-2 etc. Effectively the same integral 1000 times, and the different limits and constants balance out in the end (try it to see that).

Re: I liked this simple calculus exercise

#17

Post by mkl » Mon, Apr 17, 2023, 6:13 AM UTC

noobcoder wrote: post #5

It's easy to fall into the trap of relying on rote memorization of integration rules, but problems like (⋆) force students to truly understand the concepts behind the math.

I don't think that's true. Floor is a piecewise function, so you follow the rule for integrating piecewise functions and break it into a sum of integrals of each piece, then follow the rules for those (they're all basically the same, so you don't need to do 1000 of them). You don't need to think about periodic functions at all.

Re: I liked this simple calculus exercise

#18

Post by markisus » Mon, Apr 17, 2023, 6:24 AM UTC

saghm wrote: post #11

ryan-duve wrote: post #2

This reminds me of an exercise I'll never forget from my Math Methods course: finding the derivative of arcsin(x). It seems almost impossible because, just looking at it, there seems to be nothing you can do to simplify it. Then, out of sheer nothing-else-to-do-ism, you take the sin() of it and realize sin(arcsin(x)) = x. Take the derivative of both sides, apply chain rule and draw a right triangle and you have the a…

In turn, this reminds me a bit of a calculus problem I saw freshman year of college that I ended up sharing with my old high school calculus teacher because it similarly looks intractable until you make a "simplification". The problem was to find the integral of `x/(x + 1)` (I forget the exact bounds, presumably from 0 to x with respect to x). The trick is that this the same as `(x + 1 - 1)/(x + 1)`, which you can th…

Why not directly u substitute u = x+1

Re: I liked this simple calculus exercise

#19

Post by chii » Mon, Apr 17, 2023, 6:37 AM UTC

mkl wrote: post #17

noobcoder wrote: post #5

It's easy to fall into the trap of relying on rote memorization of integration rules, but problems like (⋆) force students to truly understand the concepts behind the math.

I don't think that's true. Floor is a piecewise function, so you follow the rule for integrating piecewise functions and break it into a sum of integrals of each piece, then follow the rules for those (they're all basically the same, so you don't need to do 1000 of them). You don't need to think about periodic functions at all.

> they're all basically the same

> You don't need to think about periodic functions at all.

except you just described that which is called period...so it's actually good for a student to notice these things, and use the correct term, so that they can associate the name with the idea.

Re: I liked this simple calculus exercise

#20

Post by calf » Mon, Apr 17, 2023, 6:54 AM UTC

arjie wrote: post #14

Ha 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…

Floor functions trigger my fear instinct, but at least with this question I could just sit and visualize what the graph looks like, e.g. the basic sawtooth function from first year engineering.