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Math Help - There doesn't seem to be much to calculus.

  1. #1
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    There doesn't seem to be much to calculus.

    I learned calculus from Kumon, which is a place to learn math. I admit university will teach more than Kumon (obviously), but Kumon teaches most of what university teaches, mostly lacking in application (they're restrained on number of worksheets)

    It didn't take Kumon many pages to teach math from counting numbers to calculus (4620 sheets size of half of A4 to be exact. Also, Kumon already used 2000 sheets before the introduction of fractions). I remember it took Kumon 1 page to teach how and the logic behind how differentiation works, 5 pages to teach all the differentiation methods (except inverse trigonometric functions, which they don't teach), 10 pages to teach all the applications of differential calculus. 10 pages to show how integration works and all the integration methods, including (a few methods of) differential equations. 1 page was used to teach Riemann sums, how they work and how to convert limit to integral, 10 pages to teach many of the application of integrals.

    Is there all there is to calculus? If that's it, how do you fill multiple long lectures? With countless problems?

    BTW, the number of pages is a close approximation (don't have time to count them). Also, I'm assuming each page is full of instructions, because some instructions only take half a side, and never exceed 1 side, so after cut and paste, I get the number of pages.
    Last edited by chengbin; July 4th 2009 at 08:32 PM.
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  2. #2
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    you fill lectures with theory and interesting application questions, but if you just accept things and take calculus at face value and memorize, there isn't much to it

    if you try to prove stuff, it's a lot more interesting

    on a more advanced note, the inner working of what's happening and why calculus works in trivialized in calculus classes, it requires analysis which is generally a high level class at universities
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  3. #3
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    I like what I've been hearing about Kumon so far.

    Calculus is sometimes defined as the study of limits. Once you have mastered the definition of limit, the rest of calculus -- differentiation, integration, infinite series -- follows. Slopes approach values, areas fill at certain rates, and sequences converge.

    It was fairly difficult to teach myself calculus, but looking back on it now, I realize that many of its concepts were deceptively simple. Over time, I have gained so much respect for it that it is now one of my most favorite subjects.
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  4. #4
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    Oh, I forgot to mention that the calculus taught in Kumon is ONLY single variable, so no double or triple integrals.

    Kumon actually do prove most of the formulas and theorems (they even proved most of the basic area and volume formulas when we hit calculus). Actually the painful thing about Kumon is by the time we hit algebra, we had to figure out a lot of the formulas and theorems out ourselves (with hints of course). I think the worst was when we learned trigonometric substitution. We got a question, gave us the hint to use \sin \theta, then 2 questions later,

    \int \frac {dx}{3+x^2}

    No hint. We had to magically know to use \sqrt 3\tan \theta

    Then a few pages later

    \int \frac {dx}{1-x+x^2}

    No hint either, we had to magically know to complete the square.

    Then a few questions later.

    \int_0^4 \frac {dx}{\sqrt {x^2+9}}

    Which we don't know that you need to do a special substitution to integrate.

    But yeah, this was going into absurd length to save pages. Good thing I did that part in class, and the "professor" explained it to me, otherwise I would not have understood trig substitution. I can name another one, like partial fractions, where we didn't learn fraction decomposition (actually we did, but there was 1 page on that several levels ago, and we didn't know what happens when the denominator is raised to a power)
    Last edited by chengbin; July 5th 2009 at 06:52 AM.
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