So, as an addition to the How do YOU Study? thread, here is a thread where you can share your best mathematical secrets. What makes you good at calculus? What gives you an extra edge in algebra? What are your secret tricks?

Here is one of mine. It might not seem like much, but it's pretty good. If you all behave, you might get some more. When I'm attempting to solve a problem, I only write after I've thought long enough. This might seem obvious, but many, many people set out to solve problems and begin writing right away - this is like talking without thinking, only worse. When I break that rule, it's never pretty - it either results in a few pages of worthless scribble, in major depression, or in both.

Use subconscious mind and lucid dreaming to solve problems.

I always re-write the problem first so I understand what it is asking...but on the whole, I would agree.

My "secret" would be if you are doing a lecture course always answer the tutorial questions. Always. If there are any you can't do: ask! Similarly, if you are reading a book do the questions! You can't help but understand if you are forced to think about stuff! (Although I am unsure if this theory holds for Analysis...)

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Like Swlabr, I always rewrite the problem. I guess a teacher once told us to do so in an exam, and ever since, I've adopted this method. And it's pretty useful.
The same way as I said in the topic "How do you study", I'd say it's a lot better if you've written the important thing yourself at least once.

Then I put the main ideas that can be directly taken from the text. And then think by writing, because I'm always messy, but I like to have everything just around.

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Everything is possible. The impossible just takes longer.


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I write down what I know first - such as non-given constants/equations etc...

I also learn the basics and derive any complicated equations from the basic one

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Similar to above.

Learn what formulas you can derive and those you can't. When I was in Calculus I and II there were tons of things to memorize but many of them could be reduced. Some things like Simpson's Rule or I just memorized the coefficients to save time. A great example of reducing memorizing is trig identities. Start with and the other two forms follow by dividing by sin(x) or cos(x). You all know that for sure but it was painful watching classmates memorize all three forms separately. Sometimes you can derive but it takes too long. Example: quadratic formula.

In algebra II I would use my calculator to test rules I wasn't sure about to double check. If anyone thinks about something like for a second it becomes obvious it's false.

Finally on AP math tests (or free response in general) if I couldn't get part A or a 5 part question and the next parts depended on the answer for A I would just pick something like 1. You can end up not knowing the idea of the problem at all but get most of the credit by thinking smart.

I wouldn't say the following makes me good at calculus but it definitely makes me much faster....


I try to avoid u-substitution at all costs when integrating. By this I mean I almost never actually write down let u= and then subsitute it back in so that I can see an obvious integration pattern.

I firmly believe that when learning calculus you should utilize and guess and test method in your head, and not write down u-subs. I've been integrating this way from the beginning, due to a fantastic teacher, and at the University level I was finishing my calc tests 25 minutes into the class. My professors were shocked and one even asked me where my u-sub was. I replied I don't need it, cuz there's your answer


Example:

You should know that the answer is going to involve
Moreover, it's going to involve because there's no other way to get the 5 in there

Now take the derivative in your head, you'll realize that you'll be off by a factor of 5, so stick a in front of your initial guess.


Obviously, I'm just not writing down the u-sub, but that's because you simply don't need to

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Using elementary concubinary logic, you can easily show that the flumex is both semi-dependent and, even more importantly, quasi-invariant

Some can do math in their sleep, and other will burn all midnight oil and still fail. A gift makes a whole world of a difference.

Why do engineers say they hate math? Hmmm!

I understand avoiding unnecessary writing to save time but I think the idea presented here is one that could hurt you more than help you. The example you gave is simple enough to do in your head, sure. Each intermediate step that is skipped on paper and done in your head adds more and more likelihood of an error in my opinion. Also one of the students I tutor constantly makes all kinds of errors from too skipping steps on paper so this makes me frustrated on the topic in general .

I'm curious, do you never use u-substitution when integrating or just for basic problems? Where do you break down and use it if so? What about for other substitutions like trig sub?

Jameson, you made me feel better. When I heard him saying there isn't any need for the u sub, I felt very ill thinking I am too dumb.

In my experience tutoring calc, kids normally complain about being pressed for time, and then do simply integrals like the one above with u-sub, which drives me crazy.

I do write down trig subs and hyperbolic trig subs, but thats mainly because I don't encounter them too often (especially these days)

If I cannot get an integral after maybe 2 minutes of pure thought I might start fooling around with u-subs, but I really dont usually have to

For integration by parts I go straight to the tabular method

Normally my only u-subs come from problems with radicals in both numerator and denominator with different radicands, or something like that



If you truly have differentiation down, I feel "guessing and testing" would be quicker for many people who just weren't introduced to it

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Using elementary concubinary logic, you can easily show that the flumex is both semi-dependent and, even more importantly, quasi-invariant

I figured you did something like this. I agree that after a long time of doing certain types of problems you can get a feel for when it's time to write out all the steps and when it isn't. Many students think they reach that level too early though but that's what keeps me in business tutoring

I will say that I didnt use u-sub right from the get go. That's how my teacher taught it and looking back I couldnt be grateful enough. We later learned ths u-sub business. I was introduced to guess and test and since I, and the rest of my class, was so familiar with differentiation, it was no problem at all

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Using elementary concubinary logic, you can easily show that the flumex is both semi-dependent and, even more importantly, quasi-invariant

Calculus just clicked with me. In high school I never studied for AP Calculus and I got a 100 in the class and a 5 on the AP exam. In college, I study, but not to the extent I know some students do. For me, I guess it's a combination of the subject just clicking with me and a diligence of doing the homework problems. Also, as I stated in the study thread, I always try to reword theorems, etc. in my own words and put the material into a context that makes sense to me. Sometimes textbook language can be quite dry, but I am pretty good at changing such dry language into a vivid context that is coherent and logical to me.

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Everything under the sun is in tune, but the sun is eclipsed by the moon.

Like the above poster said, I also have the ability at times, to solve problems in my sleep