A High-School Freshman's Journey Into Calculus

The short version: What path should I take to learn Calculus in High-School?

The long version: I am a high-school freshman, and I am enamored with mathematics, and I just see beauty in many things when it comes to math. One of which is derivatives. For the past week or so, I have shifted my focus from my regular math class (geometry, ugh) to the beginnings of Calculus. And I am in love with what I have seen so far.

Before I go any further, I would like to give a little proof that I am not making this up: I formulated the Power Rule all on my own, just by looking in the patterns between the base equation and the derivative of it. My math teacher just about had a heart attack.

So here is my question: What is the order in which you guys (and maybe girls :) would suggest that I go through in Calculus? So far, I grasp a fairly solid knowledge of the Power rule and Limits, and I can find the derivative of just about any (sorta simple, nothing too complex ;) equation thrown at me. Currently, I am using Kahn Academy for my main resource. So, anybody have some input? :)

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Sure, how would use these patterns to differentiate $2^x$? I think your pattern matching will not work for this case. We look at the limit of the difference quotient for a good reason. It is true that the rule $\frac{d}{dx}x^n = n x^{n-1}$ is simple to understand, but that is not the same as understanding differentiation. It's deeper than that. – James S. Cook Nov 8 '12 at 18:45
With the $2^{x}$, I believe that $d/dx 2^{x}$ would equal $2x^{x-1}$ (I am not good in TeX, how do you make the derivative thingy?) – fr00ty_l00ps Nov 8 '12 at 18:54
Always better to ask the question at the top, so people can quickly find out if they have any interest in trying to answer. By burying your question at the end of a lot of personal stuff, readers have no idea until the end what kind of question you are asking. – Thomas Andrews Nov 8 '12 at 19:13
Does your high school have a calculus class? If so, may I recommend you teach yourself something else? Calculus is something you can pick up anywhere, and it is not actually the best place topic to learn to think in a sophisticated way about mathematics - you either "gloss over" the details or the details get very hairy quickly. Number theory is a better place to start maturing as a mathematical mind, IMHO. – Thomas Andrews Nov 8 '12 at 19:28
I second the suggestion on number theory. Discrete mathematics in general is a rich subject which is not covered adequately in the high school curriculum. It's one of the few subjects where very little technical background is needed and you can simply jump in. – EuYu Nov 8 '12 at 19:44

1 Answer

You might very well want to visit the art of problem solving's "online school", where you will find both curriculum with which to proceed in learning calculus (and more!), and where, perhaps, you'll find a sense of community with other young math enthusiasts!

For a free and credible "text", see MIT OCW Calculus online text. What's nice is that each section/chapter is available for downloading in pdf.

ADDED: See also S.O.S. Calculus.

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I actually very much prefer learning at my own pace and on my own, jumping to one thing from the next. Although this strategy will not be a complete learning solution, but it seems it will be the most fun for me, to keep me interested ;) – fr00ty_l00ps Nov 8 '12 at 19:13
I believe you can learn at your post with the AOPS online school. When I referred to "curriculum" - I meant you'll have access to what you need to learn calculus. When I referred to "community" I meant that, through the use of discussion forums there, you can communicate and/or socialize with other eager students. – amWhy Nov 8 '12 at 19:16
Well, I am on a budget of absolutely nothing. I have just a little bit of money at my disposal at the moment, all of which is going towards a new computer to do large calculations on... So I would think to turn to MIT Open Courseware, but I would rather be able to be spontaneous and jump around ;) – fr00ty_l00ps Nov 8 '12 at 19:21
You're a bit inconsistent here: you asked if there are any recommendations for paths you might take to learn calculus, and asked in what order you should cover material, and yet you've said twice here that you prefer to be spontaneous and jump around? It needn't be one or the other. – amWhy Nov 8 '12 at 19:25
Ah, true, I just want to keep myself going in a general direction, making sure I don't completely miss something ^^ – fr00ty_l00ps Nov 8 '12 at 19:26