Math

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Interesting thought... in your equation, can you give an idea of what the graph of "X" against "x" might look like? (For various values of N)

That depends on which calculus course you take. Some of them are more rigorous on understanding and will test you by asking proving questions. If you want my opinion of it, I prefer courses that have an equal compromise between computations and proving (the understanding bit). Besides, if you understand it, you have no need to memorize it; you will know it simply by deriving it. Of course, for a test situation, you might not have time to derive the equations again, and might be forced to memorize it. It might be more accurate to say that you might only use calculators for calculations that are difficult to do in your head. Because 4000000 is a huge number, but multiplying it by another huge number with about the same number of zeroes is not really all that difficult, and shouldn't require a calculator, anymore than you need a calculator to know what 1+1 is. I sincerely despise using a calculator because it makes you rely on it too much. You might encounter a question you don't know how to answer because you don't know how to solve it, and no calculator in the world could help you do that. If you rely on a calculator too much, you might start to believe that you couldn't solve the problem because your calculator wasn't good enough.

Case in point? Consider trying to figure out whether or not 3^5555 + 5^3333 is divisible by 7. If you rely on the calculator too much, you'll punch in the numbers and divide it by 7 to see if you get a decimal point. Then your pocket calculator, having only a maximum of 8 digits, doesn't process such large numbers. So you believe you need a better calculator. But if you don't rely on the calculator, you soon realize that 3^6-1 is divisible by 7, and 3 to any power divisible by 6 divided by 7 always results in a remainder of one (this requires use of the mod concept, but once you understand it, it's not usually difficult to apply), and since 3^5555 = (3^6*925)(3^5), the remainder of 3^5555 when divided by 7 is 1*5 = 5. Continuing in this fashion, you can easily figure out what the remainder is for 5^3333, add the two remainders, and see if you get 7. Which proves that you don't need a calculator to do this kind of question. And for the record, no, I did not use a calculator to figure out whether or not 6 divides 5555 or what the number closest to it was. This kind of mental math can be done quite easily in your head, although I'm not saying this to toot my horn; I'm saying this because it is possible. I am not a human calculator. There are people out there even more extraordinary than myself at that. I know a few of them.

Of course, I consistently like to prove my point by taking my math tests (some of which allow pocket calculators) without any calculators, and do some of the longer or bigger calculations by hand, which isn't much anyways, and complete the test before the end of the period, and get a relatively high score on it--proof that even if we are allowed to use a calculator for a test that might be designed for such purposes, you need not have one to ace it.

You might also be interested to know that most professors or teachers won't dock marks if you don't arrive at a completely refined answer. 32*468 is as good an answer as 14976.

I would like to think that math shouldn't be about memorizing, but more about understanding it. And this is why I prefer it to dry subjects like history or some sciences. If you can understand how a math proof or theorem works, you simply don't have to memorize it--it will come to you when you go through the proof or theorem again.

However, you usually don't have time to do this on a test, so you might end up memorizing the proofs or their results so that you can solve the problems, but by the time you get around to doing the test, you should have done so many practice or assignment problems that doing it would be like second nature to you anyways. Of course, doing practice problems is the only way to get better at math. You can prefer not to attend any lectures and still ace a math course. Not that I'm recommending that method though; that depends on how you prefer to study.