I liked this simple calculus exercise

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 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&#x27;m not sure if it&#x27;s how I was taught, how I studied it or something else but calculus always seemed like a huge step change in difficulty.<p>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.

This reminds me of an exercise I&#x27;ll never forget from my Math Methods course: finding the derivative of arcsin(x).<p>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.<p>Like the words the author uses for the integral, it&#x27;s all valuable technique.

Current Calc 2 student here. I would be braindead approaching this problem honestly, I don&#x27;t think I&#x27;d even know how to begin; I&#x27;m hoping that&#x27;s normal.<p>Why would the exponent be equal to x&#x2F;2 - floor(x&#x2F;2) be equal to x&#x2F;2 on the interval [0, 2)? And how does the graph of x&#x2F;2 - floor(x&#x2F;2) imply anything about the behavior of e^(x&#x2F;2 - floor(x&#x2F;2))? I&#x27;m hoping I just haven&#x27;t learned enough yet?

Reminds me of college when I said to my Real Analysis professor &quot;that&#x27;s a neat trick&quot;. His response: &quot;It&#x27;s not a trick, it&#x27;s a method.&quot; :-)

First rule of integrating non-analytic functions: If they&#x27;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!<p>(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.<p>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.<p>Here&#x27;s someone writing it out on video on a tutoring site <a href="https:&#x2F;&#x2F;www.doubtnut.com&#x2F;question-answer&#x2F;int050exdx-where-x-denotes-the-fractional-part-of-x-630440021" rel="nofollow">https:&#x2F;&#x2F;www.doubtnut.com&#x2F;question-answer&#x2F;int050exdx-where-x-...</a><p>You have to spot the period, but x - floor(x) is called &quot;fractional part of x&quot; 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.

Slightly off-topic: didn&#x27;t know what ⌊x&#x2F;2⌋ is<p>Google: x squared (???)<p>GPT: The expression ⌊x&#x2F;2⌋ represents the greatest integer that is less than or equal to x&#x2F;2. It is called the floor function of x&#x2F;2. For example, if x=5, then ⌊x&#x2F;2⌋ = ⌊5&#x2F;2⌋ = 2. If x is an even integer, then ⌊x&#x2F;2⌋ = x&#x2F;2. If x is an odd integer, then ⌊x&#x2F;2⌋ = (x-1)&#x2F;2.

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.<p>After watching Michael Penns youtube channel [1] for some time now, and he <i>loves</i> 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.<p>[1] <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;@MichaelPennMath">https:&#x2F;&#x2F;www.youtube.com&#x2F;@MichaelPennMath</a>

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&#x27;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.<p>I&#x27;m actually looking for a set of such problems as I think it&#x27;s a lot better than grinding out hundreds of quadratics or polynomial derivatives and such. I found the AOSP stuff already, wonder if there&#x27;s other good sources.

We used to get lot of such tricky stuff during the preparation of IIT-JEE here in India, and I&#x27;m telling you if you don&#x27;t understand Area under curve is integral, you can&#x27;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&#x27;ll love them

What does the [x&#x2F;2] notation mean, here?

It&#x27;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.

As with so many integration exercises, this seems more like an exercise not in integrating, but in your understanding of the relevant functions.

Nowadays I use Wolfram Alpha for stuff like this.

&gt; If some expression looks complicated, try graphing it and see if you get any insight into how it behaves.<p>This is not always a good idea. Some functions have complicated behavior that makes them either plain hard to draw (e.g. sin(1&#x2F;x) near 0), or reach very high values but also be near 0, or be otherwise tricky.

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&#x27;t always able to solve those, but it was so satisfying when I did (usually after an hour or two of struggle).

I am a visual person. A good strategy for me is to plot the integrand to gain insight to the problem.

That technique won&#x27;t scale if you&#x27;re trying to integrate some crazy complex function.

This would make for a good mental math interview problem. Thanks.

I think this exercise is dumb. Not interesting, not challenging, not useful, not anything. I&#x27;d feel offended, if someone actually approached me with it.