[HN Gopher] I liked this simple calculus exercise
___________________________________________________________________
I liked this simple calculus exercise
Author : kamaraju
Score : 438 points
Date : 2023-04-17 01:24 UTC (21 hours ago)
(HTM) web link (blog.plover.com)
(TXT) w3m dump (blog.plover.com)
| tachyon5 wrote:
| Great 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.
| kolbe wrote:
| This would make for a good mental math interview problem. Thanks.
| ryan-duve wrote:
| 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 answer.
|
| Like the words the author uses for the integral, it's all
| valuable technique.
| mjd wrote:
| This is a great example. Thanks very much!
| justeleblanc wrote:
| That's how you find any reciprocal function's derivative. If g
| = f^{-1}, then g'(y) = 1/f'(g(y)).
| mydogcanpurr wrote:
| The approach is less mysterious if you recognize it as going
| back to definitions, which is a common problem solving
| technique.
| 77pt77 wrote:
| Dude. That's just a direct application of the inverse function
| theorem.
|
| Even geometrically you can see that swapping the axis (x and y)
| gives you the desired result.
| CGamesPlay wrote:
| GP just happened to be one of the lucky 10,000 on that day in
| math class. https://xkcd.com/1053/
| tonyarkles wrote:
| Alternatively, x = sin(theta) and arcsin(sin(theta)) = theta,
| depending on the limits. It's a cool technique, thanks for the
| reminder!
| ogogmad wrote:
| One technique for finding the derivative of the sin function is
| to find the derivative of arcsin first. The arcsin function can
| be expressed as the area of a certain figure, and therefore
| admits an expression as an integral. From this, the derivative
| of arcsin is immediate. Finally, apply your technique to
| arcsin(sin(x)) = x and obtain the derivative of sin.
|
| A similar technique finds the derivative of exp(x) from ln(x),
| by defining the latter as the integral of 1/x.
| selimthegrim wrote:
| ChatGPT?
| ogogmad wrote:
| :P See https://math.stackexchange.com/questions/75130/how-
| to-prove-...
|
| The OP uses the derivative of sin to find the derivative of
| arcsin. My point was that the other way round is sort of
| natural as well. Thought I'd share.
|
| The broader point is that you can obtain all the classic
| definitions and identities of the _elementary functions_
| [1] by combining only (i) the four arithmetic operations,
| (ii) integration, and (iii) taking inverses of functions
| (iv) the constants 0 and 1. I guess that 's a bit cool.
|
| This works fine over the real numbers. Over the complex
| numbers, this might require Riemann surfaces to work. e^z
| in complex analysis is usually defined by its Taylor
| series, instead of as the inverse function of the natural
| log (whose domain is actually a Riemann surface).
|
| [1] - https://en.wikipedia.org/wiki/Elementary_function
| saghm wrote:
| 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.
| markisus wrote:
| Why not directly u substitute u = x+1
| ketzu wrote:
| The thing I dislike about many maths problems (including many
| proposed in this thread) is taking the wrong initial approach can
| make it take forever. Finding the right trick to solve something
| can feel enlightening, but in my experience it feels mostly
| frustrating if you waste 10x the time by taking one wrong step in
| the beginning.
|
| After watching Michael Penns youtube channel [1] for some time
| now, and he _loves_ the floor function, I recognized what was
| going on - and wondered how I could prove this is 1000 times the
| simple function beyond just stating it.
|
| [1] https://www.youtube.com/@MichaelPennMath
| nateb2022 wrote:
| 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?
| Scarblac wrote:
| 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".
| hgsgm wrote:
| 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.
| monoprotic wrote:
| > Why would the exponent be equal to x/2 - floor(x/2) be equal
| to x/2 on the interval [0, 2)?
|
| floor(x/2) = 0 on the interval [0, 2), so the expression
| reduces to x/2.
|
| > And how does the graph of x/2 - floor(x/2) imply anything
| about the behavior of e^(x/2 - floor(x/2))?
|
| If y = x/2 - floor(x/2) is periodic, then e^y = e^(x/2 -
| floor(x/2)) must be periodic as well, with the same period.
| nateb2022 wrote:
| Thank you!
| actionfromafar wrote:
| 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.
| mjd wrote:
| 1. Look at the graph! You can _see_ that the graph, between 0
| and 2, is a straight line through the origin.
|
| 2. On the interval (0,2), the expression x/2 is a number less
| than 1. The floor of this number is zero.
| itengelhardt wrote:
| 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.
| tgv wrote:
| Surprising how many people have never seen floor/ceiling
| notation.
| jrumbut wrote:
| 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.
| arbitrandomuser wrote:
| > 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. Also even with the example in the
| post , thinking about the problem at first and exploiting
| it's periodicity before naively integrating numerically over
| the entire limit affords much faster computation. Although
| might not be specific to this example , utilising such
| symmetries can sometimes turn even a numerically intractable
| problem or a problem that requires days / 100s of gigabytes
| into something one can do on their laptop in hours.
| pastaguy1 wrote:
| calc 2 was the hardest for me, and anecdotally many others.
| hang in there!
| shortrounddev wrote:
| > I'm hoping that's normal
|
| The vast majority of people on the planet do not know calculus
| and will never need to, so yes it is completely normal
| mkl wrote:
| 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).
| echojc wrote:
| 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 couple of
| minutes to plot this on a graph for a small range of values,
| like 0 <= x <= 6 (deciding what range to check is also part of
| your problem solving skillset).
|
| With this, you can calculate and sketch out e^(above result) on
| a graph. Finally, knowing the principle that a definite
| integral calculates the area under the curve, you should be
| able to use your sketch to reason out how to calculate the
| entire original integral.
|
| Hopefully you can see how solving this kind of problem isn't
| about knowing anything about this particular problem, but
| simply investigating it without any prior expectations, which
| is why the author thinks this is an interesting exercise for
| students.
| hgsgm wrote:
| 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).
| mjd wrote:
| > All you need is to notice...
|
| All you need is a magical inspiration from out of nowhere!
|
| But if you don't happen to have that magical inspiration,
| the graph will make the periodicity visually obvious.
| lordnacho wrote:
| 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.
| apocadam wrote:
| Here's one such place -
| http://www.math.toronto.edu/oz/turgor/archives.php
| einpoklum wrote:
| > 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.
| mjd wrote:
| There is nothing that is always a good idea.
| arjie wrote:
| 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.
| planede wrote:
| That's just an algebraic application of the same idea. I don't
| think it's sad.
| calf wrote:
| 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.
| de6u99er wrote:
| Nowadays I use Wolfram Alpha for stuff like this.
| keithalewis wrote:
| That and ChatGPT got yer Rightin' an 'Rithmatic covered. Watcha
| gonna do 'bout yer Readin'? Then we ain't need no schoolin'
| nh23423fefe wrote:
| Living in the past suits you.
| cperciva wrote:
| 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.)
| red_admiral wrote:
| Here'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.
| amelius wrote:
| The problem is that most teachers cannot come up with questions
| that go outside the small number of cases which the student
| trains on. I mean, coming up with fundamentally new questions
| is very hard work once the low hanging fruit is gone.
| red_admiral wrote:
| That or the point of the class is to prepare for a test set
| by someone other than the teacher (such as in UK A-levels),
| where you know the questions will mostly be of the form
| "derivative of 3x^2-5x+1" or similar.
| AlecSchueler wrote:
| Sounds like something ChatGPT could do quite quickly though.
| petschge wrote:
| Quickly maybe. But to do it well you need a deep
| understanding of the material which is something that large
| language models lack.
| whoami_-h wrote:
| [dead]
| pavlov wrote:
| I learned basics of calculus as a teenager making animations
| with a pirated copy of Adobe After Effects. It's a motion
| graphics package where you can animate any property of an
| element over time using both absolute values and velocity
| curves (i.e. the derivative). It shows both curves next to each
| other, so you can tweak either and see how the other changes.
|
| Seeing graphics animate according to a derivative that you just
| plotted yourself is really useful to develop practical
| intuition about what it means.
|
| After Effects is too expensive and complex for high schools,
| but maybe some kind of modern Logo-style environment that
| combines coding and animation could be useful for calculus
| beginners. (And linear algebra too -- another field where the
| basics have a direct intuitive application in computer
| graphics.)
| mrguyorama wrote:
| Geogebra and friends are more powerful and open-ended,
| education focused systems. I played around with it a bit in
| freshman year Geometry.
| irchans wrote:
| This is a good idea. If I ever teach basic calc again, I will
| use it. Thanks!
|
| (The last time I taught math was vector calc 4 years ago. The
| last time I taught the first semester intro to calc was 30
| years ago OMG-LOL.)
| angry_moose wrote:
| I remember there being a distinct lack of "closing the loop" on
| concepts in university math.
|
| Day 1 of the class: The derivative calculates the slope of a
| function
|
| Day 2: The integral calculates the area under the curve of the
| function
|
| Days 3-89: Rote exercises deriving and integrating increasingly
| obscure functions
|
| Day 90: Final Exam
|
| Spending a few days at the end re-exploring the "big picture
| day-1" to tie together all of the various strands of knowledge
| you accumulate over the semester would have made all of it so
| much more effective.
| nerdponx wrote:
| This, bigtime. I loved the 3blue1brown series an example of
| how to do this better.
| sizzle wrote:
| I wish I could give this comment hold, straight up put me off
| math because of this lack of real world application in lower
| level calc and adhd brain craving meaning and substance.
| sizzle wrote:
| *Gold
| deanCommie wrote:
| It's interesting because I was going to argue with you until
| I applied the same idea to a subject I couldn't care less
| about if I tried in school, but now think is deeply
| fascinating: History.
|
| See, I loved math. So all through Calculus I could easily
| remember the big picture. Every time I practiced an Integral
| I imagined curves and calculating the areas under them and
| the visual problem and the relevance of what the curve
| represented in real life and the massive amount of
| applications it could be used for in the real world lit up my
| neurons like fireworks in the sky.
|
| (Then I went into computer science, and wound up never
| needing calclus again, based on the type of work I happen to
| be doing, but alas)
|
| But where I really would have wished a constant reinforcement
| of the "big picture" is History. Cuz it always seemed so
| pointless and useless. Why are we studying these old dead
| people, and everything they did. Who cares? They're old, and
| they're dead, and nothing they did matters to us anymore.
|
| Until you grow up, and go from your 20s to your 40s, and
| suddenly realize oh shit we're living THROUGH history. We're
| creating history NOW. We're making choices, and we're making
| _mistakes_ just like those old dead people in history. Old
| dead people that weren 't really any less developed or
| evolved primates than us. Just equally victims of their
| circumstance like us, and also agents of change like us.
|
| Suddenly history seems much more significant.
| postsantum wrote:
| Funny enough, I ended up with the opposing view - history
| doesn't matter, narratives do. During the last year I
| learned about propaganda, psy ops and myths much more than
| I knew before. And at the risk of sounding as an edgy
| teenager, I can vouch for the phrase "History is written by
| victors" which stopped being a platitude for me and became
| a part of my worldview
| deanCommie wrote:
| You're not wrong but what's the difference between
| history and narratives?
|
| "History" is the study of different narratives to TRY to
| come to a semblance of truth, but even for recent events
| this is almost impossible.
|
| Pick an example like "Did the US dropping the nuclear
| bomb on Japan ultimately save lives, or was it
| unnecessary" and it's impossible to find the truth
| between 2 conflicting narratives, each fairly
| justifiable.
|
| Another one is, most Soviet Anti-American Propaganda was
| true.
| BeFlatXIII wrote:
| I prefer the existentialist "history is written by those
| who write history." Often, the victors give themselves a
| better justification for their conquest than admitting
| that someone bigger and badder may eventually dethrone
| them. However, some losers write histories praising their
| new overlords and their incredible military might, not
| out to gain favor with the new king but to preserve the
| dignity of the old king. The defeat was inevitable
| because of the invader's futuristic military powers aided
| by the Gods (instead of admitting the old king had no
| strategy).
| andrewflnr wrote:
| I think history is potentially even easier. Everyone asks
| "why?" about things being the way they are, or can be
| interested in it if you remind them that things could be
| otherwise. History is the answer to "why?" for a huge
| number of things with both small and huge life and death
| consequences. Put another way, "why" is a good framing for
| a lot of big picture history stuff.
| andruby wrote:
| My school teacher asked us this exact exercise several times.
| He always made sure to link abstract concepts to real
| applications, as well as showing us how the mathematical
| concepts were discovered.
|
| (note: this was not in the US, but in the early 2000's in a
| small European country)
| xnorswap wrote:
| Also a good question to test intuition is being asked to sketch
| e^sin(x) and sin(e^x).
| h4ch1 wrote:
| I really enjoy https://graphtoy.com/ by the great Inigo
| Quilez to quickly verify my mental models for mathematical
| equations
| Timon3 wrote:
| Thank you for sharing that! I usually go with Desmos, but
| for a quick plot this seems even nicer, especially given
| the time coordinate!
| anthk wrote:
| TCL, package require math, a few lines of puts writting to a
| file and then gnuplot that.
| yberreby wrote:
| How does that test intuition?
| EForEndeavour wrote:
| That's one of many ways to check one's answer. The whole
| point is to be able to intuit a rough sketch of what the
| graph should be, not to use software tools to build a
| pixel-perfect render of the plot.
| jhart99 wrote:
| Seems a little much for a regular calculus class. Wouldn't
| that fit better in a complex analysis class?
| jetunsaure wrote:
| No, this would be a good exercise for first-year calculus.
| It's a very reasonable question, nothing out of the
| ordinary.
| KMag wrote:
| In fact, I was taught the graphical intuition via a few
| days of these sorts of exercises, before being introduced
| to the formulas. It worked really well, at least for 9th
| grade students in an acceleration program at the
| University of Minnesota (UMTYMP).
| bheadmaster wrote:
| Intuition tells me e^sin(x) would look similar to an ordinary
| sinusiod, except its range would be between e^-1 and e, and
| its shape would not be smooth as a sinusiod. I have no idea
| what the shape would look like, and I'm a visual learner when
| it comes to mathematics.
|
| I think most of these questions are not measuring intuition
| per se, but rather has the tested person previously seen such
| functions plotted on a graph.
|
| Either that, or my mathematical intuition has got rusty from
| years of code monkeying.
| soVeryTired wrote:
| It wouldn't look too dissimilar to a standard sinusoid,
| just shifted upwards a bit. In the range [-1, 1], e^x is
| reasonably similar to 1 + x.
| madcaptenor wrote:
| You're basically right. Here's the graph:
| https://www.desmos.com/calculator/lf3xerswdy
| bheadmaster wrote:
| Ah, now it makes sense. The slope of the sine is
| "squashed" down by the x->e^x mapping.
|
| Perhaps (regularly) seeing functions plotted on graphs is
| a necessary precondition to maintain intuition :)
| ThrowawayTestr wrote:
| Sketching equations and their derivatives was how we started
| pre-calc in high school
| bluepod4 wrote:
| I recall a similar exercise on my AP Calc exam from years ago.
|
| Instead of sketching the derivative based on the graph of a
| function, we had to sketch the function based on a table of
| data which described the function as well as its first and
| second derivatives in terms of value, existence, and sign at
| various points and intervals.
| [deleted]
| postsantum wrote:
| Slightly 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.
| kolbe wrote:
| It's the floor operator. Also known as round down.
| eesmith wrote:
| 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 .
| hgsgm wrote:
| APL, famous for baffling programmers with weird notation,
| also known for making one part of math more readable :-)
| mjd wrote:
| Sorry about this. I added an explanation to the article.
| housecarpenter wrote:
| I expect Google is stripping the [?] [?] brackets out as
| punctuation in the search, so that you're effectively only
| searching for "x2", hence the "x squared" results.
| bluepod4 wrote:
| 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.
| hgsgm wrote:
| Can you name a single-variable function that software
| engineers use that can't be graphed?
|
| How would such a function be useful?
| bluepod4 wrote:
| No. I think the term "function" is overloaded here.
|
| My point was that I viewed it solely as a _programming_
| function and not a _mathematical_ function (even though it
| exists in math libraries), hence my last sentence "Of
| course, it makes sense now that I've seen it."
|
| Out of all the functions in math libraries that I've used,
| floor/ceiling are the only ones where I had this idea for
| some reason. It was obvious to me that Math.sin(x) and
| Math.abs(x) can be graphed. I've seen those graphs over and
| over again. But whenever I used the floor or ceiling
| functions, I just thought in terms of rounding up or down
| with a predetermined rule to finish whatever piece of code
| I was working on.
|
| But as others have pointed out, I have actually worked with
| the floor function as a mathematical function in several
| math classes. I have seen the graphs. They just weren't
| called floor or ceiling functions.
|
| I just never made the connection that they were the same
| and I don't recall any computer science professor or TA
| "bridging the gap" to what was learned in the math classes.
| wholinator2 wrote:
| It's also sometimes referred to as a step-function though i
| believe that name encompasses many more functions than just
| the floor
| hgsgm wrote:
| 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.
| bluepod4 wrote:
| Yeah, in college I've worked with step/unit functions in
| both my "differential equations" and "signal and systems"
| classes for sure.
|
| But it never occurred to me and it was never presented by
| professors or TAs that they were essentially the same as
| floor/ceiling functions.
|
| Or maybe I just forgot since it's been more than a decade?
| ggrelet wrote:
| What does the [x/2] notation mean, here?
| mjd wrote:
| Sorry about this, I added an explanatory note to the article.
| itengelhardt wrote:
| Thank you for asking this question. I didn't understand the
| notation either
| elseweather wrote:
| 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
| _dain_ wrote:
| _> floor(1.999) = 0_
|
| it's 1 not 0
| le_zonzon wrote:
| 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.)
| privacyking wrote:
| That technique won't scale if you're trying to integrate some
| crazy complex function.
| xigoi wrote:
| Integrating a piecewise continuous function by splitting it is
| used all the time.
| justeleblanc wrote:
| Exam questions have two wonderful property: they have an
| answer, and the person who wrote it knows that you have the
| knowledge to find the answer.
| mrguyorama wrote:
| Nope, good teachers understand when the question they wrote
| thinking to test certain concepts falls flat, because of some
| unforeseen oversight or quirk. Something being clear and
| intuitive to a math professor is not the same as something
| being clear and intuitive to the students.
|
| The best professors strike and ignore questions that
| "failed", ie didn't accurately test what they thought they
| would.
| justeleblanc wrote:
| I don't quite see how that's relevant to what I said.
| mrguyorama wrote:
| >the person who wrote it knows that you have the
| knowledge to find the answer.
|
| This isn't always true. The person who wrote it only
| knows what they taught, not what you have learned. The
| point of the test is to find what you have learned.
|
| This is just a nitpick I think.
| eesmith wrote:
| A "calculus tyro", as a general rule, is not trying to
| integrate some crazy complex function as part of a homework
| assignment.
| falcor84 wrote:
| Which part doesn't scale? From my experience, the technique of
| looking at the graph of various components in the function has
| been really useful.
| vrglvrglvrgl wrote:
| [dead]
| bmacho wrote:
| I think this exercise is dumb. Not interesting, not challenging,
| not useful, not anything. I'd feel offended, if someone actually
| approached me with it.
| lysozyme wrote:
| My introduction to calculus was "Calculus Made Easy" by Silvanus
| P. Thompson and I always liked math profs who actively worked to
| show math for what it is: useful, beautiful but not about the
| symbols or the jargon. "Any fool can calculate!" I think is what
| he says in the book.
|
| I did some math in college and when I started knowing how to
| analyze the behavior of functions (and developing the mental math
| tools to imagine what they look like without having to actually
| draw them) that's when I felt like I was kinda getting it
| signa11 wrote:
| i _loved_ that book.
|
| unpopular opinion: martin-gardner's intro really ruined the
| start of the book (for me at least). i just ignored all of it,
| and was a happy camper.
|
| other than that, i.a.maron, piskunov, g.n.berman are all
| _heavy_ but excellent texts on this beautiful subject.
| aqme28 wrote:
| As with so many integration exercises, this seems more like an
| exercise not in integrating, but in your understanding of the
| relevant functions.
| j7ake wrote:
| I am a visual person. A good strategy for me is to plot the
| integrand to gain insight to the problem.
| physicles wrote:
| When 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).
| mjd wrote:
| I just love that little five-pointed star symbol.
|
| I also like labeling formulas with playing card suits. I think
| they are easier for the reader to distinguish when they are
| looking back for the labeled formula.
|
| A while back I had the idea of marking erroneous formulas with
| a _red_ spade. I 'm going to try doing that again because it's
| hilarious.
| navels wrote:
| 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." :-)
| lordnacho wrote:
| 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.
| azalemeth wrote:
| What's your favourite example of where it doesn't work?
| Physics is full of quasi-infinistesimal quantities and I
| always like good counter examples (ideally without invoking
| something like the blamange function or similar....)
| lordnacho wrote:
| Off the top aren't there differential equations that are
| inseparable? ie you cannot just pretend x and y are x(t)
| and y(t).
| supernewton wrote:
| Somewhat silly example: given a plane in 3 dimensions
| defined by x+y+z=1, we have [?]x/[?]y * [?]y/[?]z *
| [?]z/[?]x = -1.
| contravariant wrote:
| It basically works without issue in 1d, simply because dx
| and dy can be considered modular forms and dy is exactly dx
| times the derivative dy/dx. You can even put an integral
| sign in front of them and calculate the corresponding
| integral.
|
| Where this doesn't work is if you have more than 1
| dimension. Then you need to deal with the added complexity
| of integrating modular forms and the fact that in 2D you
| don't have df = (df/dx) dx but df = (df/dx) dx + (df/dy)
| dx. The chain rule also changes into a matrix product,
| rather than a simple dz/dx = dy/dx dz/dy.
| [deleted]
| AstixAndBelix wrote:
| No, a trick is a shortcut with respect to the more tedious
| method. By definition anything that works also formally
| works, otherwise it wouldn't.... work.
| n4r9 wrote:
| In my mind, a trick is something that applies to unusual and
| specialised cases, whereas a method is something that can
| apply to a broad, well-defined class of problems.
| mjd wrote:
| George Polya writes that a method is a trick you can use more
| than once.
|
| (Although I think he ascribes this to professors who are not
| good teachers.)
| ziroshima wrote:
| This is surely a stupid question: In the article, the graph sure
| looks like a right triangle, with a base of 2 and a height of 1.
| Wouldn't the area under this curve (from 0-2) be ~1?
| mabbo wrote:
| That's a graph of `x/2 - floor(x/2)`. The question is about
| `e^(x/2 - floor(x/2))`.
| dhosek wrote:
| Ah, but that function is the input to another function and it's
| that _other_ function that's getting integrated.
| noobcoder wrote:
| 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.
| mkl wrote:
| 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.
| chii wrote:
| > 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.
| mkl wrote:
| I don't think many would notice doing it that way, as it's
| disguised by the different limits and constant factors. The
| indefinite integrals are what is basically the same on the
| surface. There's more to periodicity than different
| sections being similar on the surface, and there's
| certainly no need for them to use the correct terms.
| ajeet_nathawat wrote:
| 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.
| aashutoshrathi wrote:
| 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
| never_inline wrote:
| I am from a rural region where IIT-JEE is not that popular, and
| I liked working these physics and maths problems initially.
|
| Unfortunately the competition has become so intense you
| practically need coaching (which is expensive) and dedicate lot
| of time, at that point it becomes grunt work. There are many
| "tricks" and "shortcuts" taught in these coachings which
| doesn't exists in normal NCERT syllabus. Needless to say I
| didn't do very well.
| laurieg wrote:
| 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.
| xivzgrev wrote:
| I think it's a common feeling. Even though they are both
| "math", they feel like different skill sets.
| mjd wrote:
| I think this is really important for good teaching. It's not
| enough to show the student how to solve the problem. One needs
| to also show the student the patterns of thought that could
| have led them to the solution. And it's not enough to show how
| _someone_ could have been led to the solution, one has to show
| how _this particular_ student could have figured it out,
| knowing what they know and being who they are.
|
| I have a blog article about this in progress.
| justeleblanc wrote:
| I'm a professor in a big university in western Europe.
| Students don't want to be led to thinking of the solution.
| They want the same exercises that they did during the
| tutorial, and they want to know in advance how to solve all
| the exercises. Any attempt to digress from this is met with
| vitreous eyes.
| kaba0 wrote:
| One of my professors used to say that "even a horse can do
| derivatives. Integration is the real deal", another one said
| that you integrate by "look at it, deeply, deeply, deeply; and
| then solve it".
|
| The point is, many part of high school math is actually really
| "algorithmic". I was one of the few in my class who absolutely
| loved coordinate geometry over "normal" geometry, because I
| simply felt really comfortable with equations -- once you have
| it down, you can basically solve it, even if it is harder than
| the "notice this and that" elegant solution.
|
| Most integration problems require this intuition-based solution
| which has a certain elegance to it.
|
| It was especially humbling to me that Wolfram alpha fails most
| of the interesting calculus problems I encountered during my
| analysis classes, but after a while I managed to solve most of
| them. But it unfortunately does disappear after not using it
| for a time..
| mjd wrote:
| > "look at it, deeply, deeply, deeply; and then solve it"
|
| That's the Feynman method: write down the question, think
| really hard, then write down the answer. Only three simple
| steps!
|
| Unfortunately, some of us are not Feynman.
| mrguyorama wrote:
| I actually hit this personally, because right up UNTIL
| calculus, math was the Feynman method for me. Everything
| always just clicked, made perfect sense, and I saw the
| beauty in it, and it was great fun.
|
| Then for calculus, we learned concepts, like what a
| derivative is, and I understood that, and understood
| conceptually (as in, what everything "means" and what it
| tells you) but I could never take that concept knowledge to
| the practice problems with me.
|
| I could follow along as the professor walked us through a
| problem, showing us what heuristics helped and what
| patterns to follow and how to manipulate the functions to
| get to something that followed one of the patterns to pull
| an answer out of your ass, but I could never commute those
| heuristics and patterns to novel examples. It's weird
| because I was great at doing the exact same thing for
| physics: Taking a novel and purposely opaque problem and
| finding which pattern it corresponds to.
| jerf wrote:
| To be honest, we would probably better serve our students in
| general by presenting integration as a numerical
| approximation, doing enough symbolic stuff to demonstrate
| that some integrations can be solved that way, doing enough
| other stuff to demonstrate why there's a ton of integrations
| that have no closed-form solution with any conventional
| functions, and then moving on to more productive things
| rather than blow over a full semester grinding out
| integrations. Integration by parts is useful as a method for
| exploring the concept more deeply, and there's a few other
| such tricks to be used primarily for ensuring the concepts
| are understood better. But I'm not convinced there's a lot of
| value in all these integration tricks.
|
| Derivatives are friendly enough that I feel like a certain
| amount of grinding is justifiable, and it's justifiable as
| practice for symbolic manipulations in general. But we're
| leaving a lot of useful stuff on the table while we're
| jamming down how to integrate with trig identities and other
| such things.
|
| But it's all pie in the sky anyhow. The Curriculum Must Not
| Be Changed. The Curriculum Is Perfect. Nothing Can Be Dropped
| From The Curriculum. I don't know what miracle would have to
| be worked to get people to reconsider the curriculum from
| some sensible perspective of what students should be taught
| rather than the way that question happened to be answered
| about 100 years ago when the curriculum froze into place, but
| it probably involves the total destruction of the school
| system at this point. I can't even get people to process the
| idea that shoving incomprehensible combinations of 450-year-
| old words in what is effectively another language at children
| and telling them this is High True Art is a bad idea, what
| chance is there of prying away the utterly vital fact that
| cos(th/2) = SqRt((1 + cos(th))/2) out of The Curriculum?
|
| Maybe if colleges continue dropping the SAT and the ACT we
| can start actually fixing these curricula.
| gizmo686 wrote:
| Most integration problems require numerical methods. The ones
| with with an analytic solution are just where someone at some
| point found a way of solving them.
| gcanyon wrote:
| I'm in pretty much the same boat re: calculus, but I think a
| lot of it has to do with problems just like this. For me, early
| in my experience with calculus I _always_ looked for the
| "graph it out"/non-calculus solution. So problems like water
| leaking out of a bucket, rocket acceleration, and other
| integrals where the underlying process is in some way linear
| always fell to non-calculus-based analysis. And thus when I got
| to problems where actual calculus was required, my non-
| grounding in the basics pushed me toward rote memorization
| which (of course) didn't stick over years of non-utilization.
| mtlguitarist wrote:
| 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.
| physicles wrote:
| 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.
| kaba0 wrote:
| 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.
| wholinator2 wrote:
| 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.
| nohaydeprobleme wrote:
| This is a great comment. To add on to a point that really
| aligns with my experiences:
|
| > "Review from time to time but don't let a hard first
| couple chapters prevent you from ever learning the
| concepts."
|
| This is a very good approach, and I wish I started doing
| this earlier. Even in my university math courses, the
| professors sometimes skipped ahead to have students focus
| on a few later chapters before coming back, or told the
| class to skip several pages in the book. I also found
| that working on later exercises in a textbook would
| sometimes help me better understand concepts introduced
| in earlier chapters.
|
| Lastly--though this may not be completely relevant to
| studying mathematics--I've explicitly been taught in
| various language courses (explicitly for audio courses
| and implicitly for in-person university courses) that
| it's okay to move ahead if I know at least 80% of the
| material. The percentage may be higher for studying math
| topics, but especially for someone self-learning out of
| interest or for a specific application, it's much more
| preferable to move forward and revisit earlier exercises
| as needed, instead of quit the book. If you find yourself
| getting lost in later chapters, there is no problem with
| revisiting earlier chapters. You'd also likely be no
| worse off (possibly even better) than many undergraduates
| studying the textbook for a course for the first time.
|
| The most important thing is just to not quit the habit of
| consistent study. Perfectionism in understanding is a
| pitfall for self-directed studies, which consistency in
| studying beats every time.
| kaba0 wrote:
| Thanks! My only gripe is that I just don't know a logical
| order to follow through with math.. this is somehow less
| of a problem in other subjects.
|
| That said, I have been long thinking about a dependency
| graph for knowledge, where the nodes are great books on
| the topic.
| sn9 wrote:
| You just have to look at the university curricula for a
| few universities' math programs.
|
| Then find the syllabus for each course and look at the
| recommended textbooks.
|
| You should be able to find the best ones that way.
___________________________________________________________________
(page generated 2023-04-17 23:01 UTC)