Comments on: Why Do We Need Limits and Infinitesimals? http://betterexplained.com/articles/why-do-we-need-limits-and-infinitesimals/ Learning shouldn't hurt. Let's share the insights that made difficult ideas click. Fri, 10 Sep 2010 15:29:12 +0000 http://wordpress.org/?v=2.8.4 hourly 1 By: Kalid http://betterexplained.com/articles/why-do-we-need-limits-and-infinitesimals/#comment-307467 Kalid Fri, 23 Jul 2010 17:43:54 +0000 http://betterexplained.com/?p=380#comment-307467 @mcmlxxxvi: Glad you enjoyed it, and thanks for the comment. I too wish there was more emphasis on true understanding vs. the "let's learn enough to pass the next test" mentality. Learning the intuition may take a bit longer than memorizing in the short term, but in the long run it gives you a more flexible set of knowledge, and not to mention it's way more fun. I sometimes see grades as a curse because rather than being an indication of knowledge, they become an end in itself vs. the learning it should represent. It's very hard to test intuition -- it's a gutcheck you need to ask yourself. But with no grades there's no "incentive" (carrot or stick) -- I don't know the answer, but I too wish there was another way. @mcmlxxxvi: Glad you enjoyed it, and thanks for the comment. I too wish there was more emphasis on true understanding vs. the “let’s learn enough to pass the next test” mentality. Learning the intuition may take a bit longer than memorizing in the short term, but in the long run it gives you a more flexible set of knowledge, and not to mention it’s way more fun. I sometimes see grades as a curse because rather than being an indication of knowledge, they become an end in itself vs. the learning it should represent. It’s very hard to test intuition — it’s a gutcheck you need to ask yourself. But with no grades there’s no “incentive” (carrot or stick) — I don’t know the answer, but I too wish there was another way.

]]> By: Kalid http://betterexplained.com/articles/why-do-we-need-limits-and-infinitesimals/#comment-307466 Kalid Fri, 23 Jul 2010 17:38:41 +0000 http://betterexplained.com/?p=380#comment-307466 @Arbie: Wow, I like that functional representation of it! Yes, integrations are a general "applying" one function to another, vs. some static multiplication just to find area (area just limits our creativity/intuition I think). Ah, I should be more clear about that... the I meant the line "y = x", that is, a 45 degree line extending from the origin. So the equation y = sin(x) looks very similar to y = x for very small numbers (sin(x) extends 45 degrees from the origin when it first starts off). Hope this helps! @Arbie: Wow, I like that functional representation of it! Yes, integrations are a general “applying” one function to another, vs. some static multiplication just to find area (area just limits our creativity/intuition I think).

Ah, I should be more clear about that… the I meant the line “y = x”, that is, a 45 degree line extending from the origin. So the equation y = sin(x) looks very similar to y = x for very small numbers (sin(x) extends 45 degrees from the origin when it first starts off).

Hope this helps!

]]> By: mcmlxxxvi http://betterexplained.com/articles/why-do-we-need-limits-and-infinitesimals/#comment-307464 mcmlxxxvi Fri, 23 Jul 2010 16:04:32 +0000 http://betterexplained.com/?p=380#comment-307464 Hello, Kalid, Very well-written and descriptive. Thank you for giving me a good and pleasant read on things past and nearly forgotten! I could only wish that more people like you were teaching in high schools and universities. Around here, the tutors are often skilled in their field, but regularly and gravely fail to convey the meaning behind the definitions, theorems and proofs they teach - only the items themselves; and the educational process plummets. Arbie: —— Around 0, sin(x) looks like the line “x”. —— I believe this means the line "y = x". Thus y_1 = sin(x), y_2 = x and y1 ~= y2 for x -> 0. Hello, Kalid,
Very well-written and descriptive. Thank you for giving me a good and pleasant read on things past and nearly forgotten!

I could only wish that more people like you were teaching in high schools and universities. Around here, the tutors are often skilled in their field, but regularly and gravely fail to convey the meaning behind the definitions, theorems and proofs they teach – only the items themselves; and the educational process plummets.

Arbie:
——
Around 0, sin(x) looks like the line “x”.
——
I believe this means the line “y = x”. Thus y_1 = sin(x), y_2 = x and y1 ~= y2 for x -> 0.

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