Expert: Scotto - 7/4/2011

Question
I'm taking AP Calculus AB in high school next year, and as I'm pretty worried about it, I'd like to do some preparation over the summer if possible. I'm already periodically reviewing the skills from pre-calculus, mostly trig (the one thing I had trouble with). Which skills from precalc come up most in Calc AB? Also, are there any basic concepts I could learn on my own, without a teacher? I got my school's calc textbook from eBay, but I'm not sure if I could actually learn hard things on my own. What would you recommend for preparation?

Thanks!

Answer
You'll learn it in calculus, but there are only so many functions to deal with.

What you'll learn first is derivatives.
The derivative of a function is the slop at a point on the function.

The next thing you'll learn is integrals.
The integral of a function is the area beneath the function.
It usually has a lower limit A and an upper limit B.
Functions are usually expressed in lower case letters, so the integral is upper case.
If the function is f(x), the integral is F(x).

To start off with, you'll probably see the derivative as lim(h->0)[f(x+h) - f(x)]/h.
These are tedious and take rather long to do, but they are always used to start off.

A table of all that is needed on derivatives is given here:
http://www.math.com/tables/derivatives/tableof.htm

The only other thing to know about derivatives is the chain rule.
That is, if f(x) = f(g(x)), then f'(x) = f'(g(x))g'(x).
For example, if f(x) = (x²-2)², this is g(x) = x²-2, so f(x) = g²(x).  Thus, f'(x) = 2g(x)g'(x).
It can be seen that g'(x) = 2x.  This means f'(x) = 2(x²-2)2x = 4x(x²-2).


After doing this for quite awhile (for me it was the next term), integrals are introduced.

The integral turns out to be the opposite of the derivative.
A table of some integrals is here: http://integral-table.com

As can be seen, there are far more integrals that derivatives.

That's the two basic things to look for: derivatives and integrals.

By the way, the integral from A to B on a continuos function gives the area under the curve between A and B.  If the function is f(x), the area between A and B is F(B) - F(A).