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Calculus!!!

These are just my notes stuck up on this website. I'll come back and clarify and stick in stuff I skipped later (later, later, later... yes, I know that I have a lot I plan to do here but OH WELL)

And found on another site (source on bottom of documents) are the AB and BC calculus "bible"s!

Limits as x goes to infinity
1. Anytime the degree of the top is less than the degree of the bottom, the limit will be zero.
2. If the degree of the top equals the degree of the bottom, the limit is the coefficient of the highest power on top divided by the coefficient of the highest power in the denominator

Limits of the Sine and Cosine
1. limit as x->0 of sin x/x = 1
2. limit as x->0 of sin x = 0
3. limit as x->0 of cos x = 1

Continiuity
f(x) is continuous at point a if
1. limit as x->a of f(x) = L
2. f(a) = L

Derivative
Suppose y=f(x). The derivative of f(x), at a point x, f'(x), or dy/dx is defined as [f(x+delta x) - f(x)]/delta x. Derivative originated as being slope.

Distance as a function of time
y = f(t) where y is distance, t is time. y = f'(t) is instantaneous velocity.

Rules for Derivatives
1. If f(x) = c, f'(x) = 0 (where c is a constant)
2. If f(x) = x, f'(x) = 1
3. If f(x) = x^n, f'(x) = nx^(n-1)
4. If f(x) = cg(x), f'(x) = cg'(x)
5. If f(x) = g(x) +/- h(x), f'(x) = g'(x) +/- h'(x)
6. If f(x) = g(x)h(x), f'(x) = g(x)h'(x) + g'(x)h(x)
7. If f(x) = g(x)/h(x), f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2
8. If y = f(u) and u = g(x) then dy/dx = (dy/du) * (du/dx)

Derivatives of Trigonometric Functions
1. If f(x) = sin x, then f'(x) = cos x
2. If f(x) = cos x, then f'(x) = -sin x
3. If f(x) = tan x, then f'(x) = sec^2 x
4. If f(x) = cot x, then f'(x) = -csc^2 x
5. If f(x) = sec x, then f'(x) = tan x sec x
6. If f(x) = csc x, then f'(x) = -cot x csc x

Implicit Differentiation
Anything with x as normal, for y's as normal except leave multiplied by dy/dx (product rule if theres an x*y), solve for dy/dx.

Antiderivatives
Given a function f(x), the antiderivative is a function F(x) such that F'(x)=f(x).
1. If F'(x)=G'(x), F(x)=G(x)+C
2. If f'(x)=x^N (N != -1; if yes then logarithm), then f(x)=((x^(N+1))/(N+1))+C
3. If f'(x)=k(g'(x)), f(x)=k(g(x))+C
4. If f'(x)=g'(x)+/-h'(x), f(x)=g(x)+/-h(x)
5. If f'(x)=(u^n)(du/dx) where u is a function of x and n!=1, f(x)= ((u^(n+1))/(n+1))+C

The Definate Integral
Given the region bounded by y=f(x), x=a, x=b and y=0; by dividing into ever smaller rectangles approximation becomes more accurate.
limit as n -> infinity, all deltax ->0 of summation of f(wi)deltax from i=1 to i=n = Sab f(x) dx
(the S is substuting for the real integral symbol that i currently cant find a button for; a is on the bottom of the integral and b is on the top)
::note: If velocity=v(t), Sab v(t) dx is distance travelled from time a to time b::
1. Sab 1 dx = b - a
2. Sab cf(x) = c Sab f(x) dx
3. Sab (f(x)+/-g(x)) dx = Sab f(x) dx +/- Sab g(x) dx
4. Saa f(x) dx = 0
5. Sab f(x) dx = -Sab f(x) dx
6. The Fundamental Theorem of Integral Calculus (IMPORTANT!): Sab f(x) dx = F(b) = F(a)

Rewritten Antiderivative rules as Indefniate Integral
S x^n dx = (x^(n+1))
S [cf(x) +/- g(x)] dx = c S f(x) dx +/- S g(x) dx
S (u^N)(du/dx) dx