81–90 of 163 posts
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.
Also a good question to test intuition is being asked to sketch e^sin(x) and sin(e^x).
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.
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.)
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…
I have a blog article about this in progress.
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." :-)
(Although I think he ascribes this to professors who are not good teachers.)
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?
2. On the interval (0,2), the expression x/2 is a number less than 1. The floor of this number is zero.
Earlier quoted context omitted.
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 bre…
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.
What does the [x/2] notation mean, here?
> 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.
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 imagin…
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.
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…
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…
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.