Calculus

Calculus is basically the study of change. It was invented by 2 rivals Newton (the gravity guy) and Leibnitz.

Limits

Definition: A limit is what the value of a function approaches to as it goes to a value.

Sometimes we cannot say what f(x) is, but we know what it approaches to!

Limits of Rational Functions as they approach infinity:

Let f(x) be a rational function such that it can be expressed as P(x)/Q(x) (where P(x) and Q(x) are polynomials with coeffecients in the set of rational numbers, Q). Let P and Q be the degrees of P(x) and Q(x) respectively.

• If P < Q, then the limit of f(x) as x approaches infinity is 0.

• If P = Q, look for the coefficients of x^P and x^Q in P(x) and Q(x) respectively, and let them be called p and q. Then the limit of f(x) as x approaches infinity is p/q.

• If P > Q, look for the coefficients of x^P and x^Q in P(x) and Q(x) respectively, and let them be called p and q (again). Then the limit of f(x) as x approaches infinity is sign(p/q)*infinity, where sign(x) = x/|x| if x != 0 and 0 if x = 0.

Derivatives (aka Differential Calculus)

Derivatives are the slopes of a graph of a function at any given point. 

How do we even find the slope of a curvy function? Well, we zoom in and we have - a line! 

But how is it a curve when we zoom back out? Because the slope keeps changing! 

Now we study how to express the change in terms of the point where we measure the slope, and that's all we're doing.

That means, at any given point x (in the domain of f), the slope of f(x) at point x is f'(x).

The derivative is also written in a fancy way (dy/dx because slope is rise/run) by Leibnitz, and I really think Newton's notation (f'(x)) is easier to look at.

A few derivative values:

----- Basics -----

f(x)                                             f'(x)

c                                                0 (the slope is always 0, right?)

cx                                               c (this is also quite obvious.)

x^c                                              cx^(c-1)

----- A bit more advanced -----

c^x                                              ln(c)c^x

log_c(x)                                         1/(xln(c))

----- Trigonometry -----

(x in radians)

sin(x)                                           cos(x)

cos(x)                                           -sin(x)

tan(x)                                           sec^2(x)

arcsin(x)                                        1/sqrt(1-x^2)

arccos(x)                                        -1/sqrt(1-x^2)

arctan(x)                                        1/(1+x^2)

More rules:

(cf(x))'     = cf'(x)

(f(x)+g(x))' = f'(x)+g'(x)

(f(x)g(x))'  = f'(x)g(x)+g'(x)f(x)

(1/f(x))'    = -f'(x)/f(x)^2

----- Chain rule -----

f(g(x))'     = f'(g(x))g'(x)

With these, we can find the derivative of a lot of more complex functions.

Integrals (aka Integral Calculus)

So integrals are actually the direct opposite of derivatives!

Hang in there - I'm working on more...