How to consolidate one's knowledge of pre-calculus

[Mépris]

Over the holidays (very long ones), I intend on going through the two Calculus volumes of Apostol. However, I feel my pre-calc knowledge of math is very shaky. There's lots of little tricks I just can't see. I'm not saying I can't do harder problems, just that they're not very obvious to me.

Take this problem for instance: http://www.khanacademy.org/video/iit-jee-hairy-trig-and-algebra--part-1?playlist=IIT+JEE+Questions
At first look, I have no clue what's going. After poking around for a few minutes, I could have gotten a little further but I would have taken a longer route to square that polynomial. With a little push, I would have gotten somewhere. That's just *one* example of an exercise that I have trouble with.

Like a recent poster in the "who wants to be a mathematician?" thread, I feel like I am very, very late and I want to sort this out asap. I don't know where to begin though. I also don't have *that* much time, so my learning will have to be somewhat structured. Initially, I thought a fun way to go about this would be going through every Math Olympiad problem I come across, then moving on to the JEE questions but that seems a tad haphazard and I'm not certain if that is the most appropriate way to proceed.

Any suggestions?

 

[I like Serena]

Hi Mépris!

My recommendation (it's a little "out there"):
Visit college, pick up a couple of books that they're using, get a couple of exams with explanations (you can get those from any senior student, who's usually willing to help), and get to it!

 

[nickadams]

I'm wondering the same thing. I'm about to enter calculus and I want to have a very strong background in the topics before calculus (I was very lazy in high school). Someone recommended Lang's Basic Mathematics book so I'm hoping that covers all the bases, but other recommendations would be great.


Edit: I just found this post in the books subforum that may be useful...

Since you asked the WHY question.

Woods: Analytic Geometry & Calculus,
The first 130 pages have plenty of early 20th century style pre-calculus to shake you up.

Burnside: Theory of Equations,
Will (most likely) explain a lot of the things from your algebra classes & way more.

Chrystal: Algebra,
Will (most likely) explain a lot of the things from your algebra classes & way more.

Allendoerfer: Principles of Mathematics,
The ultimate pre-calculus book, including some calculus, more on this book http://myrtlehocklemeier.blogspot.com/2008/02/tater-says-that-cletus-knows-him-some.html.

Serge Lang: Basic Mathematics,
offers a slightly different perspective, often times better.

These books, either all of them (recommended) or some combination (at least 3) will
more than prepare you for the best calculus books I know of: Courant, Apostol & Spivak.
A bit of self-study with this discrete math book before reading those tough
calculus books would be immensely helpful as well.

All this said, you could just pick up Kline's calculus book read as far as you can if all you
want to do is read calculus stuff, from personal experience I can tell you that every time
you're going to get stuck or not be able to predict ahead of time where things are going
is only because you're missing some knowledge contained in one of the above books, so
you could also just use the above as references for when you get stuck.