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.
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.
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.)
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.
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.
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)
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.
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.
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..
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'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.
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.
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.
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.
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?
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.)
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
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.
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.
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.
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.
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 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.
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.
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.
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.
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?
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.
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
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.
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.
> 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.
[+] [-] red_admiral|2 years ago|reply
[+] [-] angry_moose|2 years ago|reply
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.
[+] [-] pavlov|2 years ago|reply
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.)
[+] [-] amelius|2 years ago|reply
[+] [-] xnorswap|2 years ago|reply
[+] [-] bluepod4|2 years ago|reply
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.
[+] [-] unknown|2 years ago|reply
[deleted]
[+] [-] andruby|2 years ago|reply
(note: this was not in the US, but in the early 2000's in a small European country)
[+] [-] Paul-Craft|2 years ago|reply
[+] [-] Spivak|2 years ago|reply
[+] [-] ThrowawayTestr|2 years ago|reply
[+] [-] laurieg|2 years ago|reply
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.
[+] [-] mtlguitarist|2 years ago|reply
[+] [-] kaba0|2 years ago|reply
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|2 years ago|reply
I have a blog article about this in progress.
[+] [-] xivzgrev|2 years ago|reply
[+] [-] gcanyon|2 years ago|reply
[+] [-] ryan-duve|2 years ago|reply
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.
[+] [-] ogogmad|2 years ago|reply
A similar technique finds the derivative of exp(x) from ln(x), by defining the latter as the integral of 1/x.
[+] [-] saghm|2 years ago|reply
[+] [-] mydogcanpurr|2 years ago|reply
[+] [-] justeleblanc|2 years ago|reply
[+] [-] nateb2022|2 years ago|reply
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?
[+] [-] navels|2 years ago|reply
[+] [-] cperciva|2 years ago|reply
(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.)
[+] [-] lysozyme|2 years ago|reply
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
[+] [-] tachyon5|2 years ago|reply
[+] [-] arjie|2 years ago|reply
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|2 years ago|reply
[+] [-] calf|2 years ago|reply
[+] [-] postsantum|2 years ago|reply
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.
[+] [-] housecarpenter|2 years ago|reply
[+] [-] bluepod4|2 years ago|reply
[+] [-] mjd|2 years ago|reply
[+] [-] kolbe|2 years ago|reply
[+] [-] ketzu|2 years ago|reply
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
[+] [-] ajeet_nathawat|2 years ago|reply
[+] [-] ziroshima|2 years ago|reply
[+] [-] mabbo|2 years ago|reply
[+] [-] dhosek|2 years ago|reply
[+] [-] lordnacho|2 years ago|reply
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.
[+] [-] aashutoshrathi|2 years ago|reply
[+] [-] never_inline|2 years ago|reply
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.
[+] [-] ggrelet|2 years ago|reply
[+] [-] noobcoder|2 years ago|reply
[+] [-] mkl|2 years ago|reply
[+] [-] aqme28|2 years ago|reply
[+] [-] de6u99er|2 years ago|reply
[+] [-] einpoklum|2 years ago|reply
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|2 years ago|reply