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I liked this simple calculus exercise

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Re: I liked this simple calculus exercise

#31

Post by mkl »

chii wrote: post #19
mkl wrote: post #17

Earlier quoted context omitted.

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.

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.

Re: I liked this simple calculus exercise

#33

Post by apocadam »

lordnacho wrote: post #26

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.

Here's one such place - http://www.math.toronto.edu/oz/turgor/archives.php

Re: I liked this simple calculus exercise

#35

Post by azalemeth »

lordnacho wrote: post #25
navels wrote: post #15

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." :-)

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.

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....)

Re: I liked this simple calculus exercise

#36

Post by AstixAndBelix »

lordnacho wrote: post #25
navels wrote: post #15

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." :-)

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.

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.

Re: I liked this simple calculus exercise

#40

Post by lysozyme »

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

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