Derivatives: Min, Max, Critical Points...

Asymptotes

Definition of a horizontal asymptote: The line y = y₀ is a "horizontal asymptote" of f(x) if and only if f(x) approaches y₀ as x approaches + or - . Definition of a vertical asymptote: The line x = x₀ is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - as x approaches x₀ from the left or from the right.
Definition of a slant asymptote: the line y = ax + b is a "slant asymptote" of f(x) if and only if lim _(x-->+/-) f(x) = ax + b.

Concavity

Definition of a concave up curve: f(x) is "concave up" at x₀ if and only if f '(x) is increasing at x₀ Definition of a concave down curve: f(x) is "concave down" at x₀ if and only if f '(x) is decreasing at x₀
The second derivative test: If f ''(x) exists at x₀ and is positive, then f ''(x) is concave up at x₀. If f ''(x₀) exists and is negative, then f(x) is concave down at x₀. If f ''(x) does not exist or is zero, then the test fails.

Critical Points

Definition of a critical point: a critical point on f(x) occurs at x₀ if and only if either f '(x₀) is zero or the derivative doesn't exist.

Extrema (Maxima and Minima)

Local (Relative) Extrema Definition of a local maxima: A function f(x) has a local maximum at x₀ if and only if there exists some interval I containing x₀ such that f(x₀) >= f(x) for all x in I.
Definition of a local minima: A function f(x) has a local minimum at x₀ if and only if there exists some interval I containing x₀ such that f(x₀) <= f(x) for all x in I.
Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.
The first derivative test for local extrema: If f(x) is increasing (f '(x) > 0) for all x in some interval (a, x₀] and f(x) is decreasing (f '(x) < 0) for all x in some interval [x₀, b), then f(x) has a local maximum at x₀. If f(x) is decreasing (f '(x) < 0) for all x in some interval (a, x₀] and f(x) is increasing (f '(x) > 0) for all x in some interval [x₀, b), then f(x) has a local minimum at x₀.
The second derivative test for local extrema: If f '(x₀) = 0 and f ''(x₀) > 0, then f(x) has a local minimum at x₀. If f '(x₀) = 0 and f ''(x₀) < 0, then f(x) has a local maximum at x₀.
Absolute Extrema
Definition of absolute maxima: y₀ is the "absolute maximum" of f(x) on I if and only if y₀ >= f(x) for all x on I.
Definition of absolute minima: y₀ is the "absolute minimum" of f(x) on I if and only if y₀ <= f(x) for all x on I.
The extreme value theorem: If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.
Occurrence of absolute maxima: If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I.
Occurrence of absolute minima: If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.
Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.
(This is a less specific form of the above.)

Increasing/Decreasing Functions

Definition of an increasing function: A function f(x) is "increasing" at a point x₀ if and only if there exists some interval I containing x₀ such that f(x₀) > f(x) for all x in I to the left of x₀ and f(x₀) < f(x) for all x in I to the right of x₀. Definition of a decreasing function: A function f(x) is "decreasing" at a point x₀ if and only if there exists some interval I containing x₀ such that f(x₀) < f(x) for all x in I to the left of x₀ and f(x₀) > f(x) for all x in I to the right of x₀.
The first derivative test: If f '(x₀) exists and is positive, then f '(x) is increasing at x₀. If f '(x) exists and is negative, then f(x) is decreasing at x₀. If f '(x₀) does not exist or is zero, then the test tells fails.

Inflection Points

Definition of an inflection point: An inflection point occurs on f(x) at x₀ if and only if f(x) has a tangent line at x₀ and there exists and interval I containing x₀ such that f(x) is concave up on one side of x₀ and concave down on the other side.

Differentiation Identities

Definitions of the Derivative:
df / dx = lim (dx -> 0) (f(x+dx) - f(x)) / dx (right sided)
df / dx = lim (dx -> 0) (f(x) - f(x-dx)) / dx (left sided)
df / dx = lim (dx -> 0) (f(x+dx) - f(x-dx)) / (2dx) (both sided)
f(t) dt = f(x) (Fundamental Theorem for Derivatives)

---

c f(x) = c
f(x) (c is a constant)
(f(x) + g(x)) = f(x) + g(x)
f(g(x)) = f(g) * g(x) (chain rule)
f(x)g(x) = f'(x)g(x) + f(x)g '(x) (product rule)
f(x)/g(x) = ( f '(x)g(x) - f(x)g '(x) ) / g^{^2}(x) (quotient rule)

---

Partial Differentiation Identities

if f( x(r,s), y(r,s) )

df / dr = df / dx * dx / DR + df / dy * dy / DR

df / ds = df / dx * dx / Ds + df / dy * dy / Ds

if f( x(r,s) )

df / DR = df / dx * dx / DR

df / Ds = df / dx * dx / Ds

directional derivative

df(x,y) / d(Xi sub a) = f1(x,y) cos(a) + f2(x,y) sin(a)

(Xi sub a) = angle counterclockwise from pos. x axis.

Table of Derivatives

Power of x.

c = 0

x = 1

xn = n x(n-1)  Proof

Exponential / Logarithmic

ex = ex  Proof

bx = bx ln(b)  Proof

ln(x) = 1/x  Proof

Trigonometric

sin x = cos x  Proof	csc x = -csc x cot x  Proof
cos x = - sin x  Proof	sec x = sec x tan x  Proof
tan x = sec2 x  Proof	cot x = - csc2 x  Proof

Inverse Trigonometric

arcsin x  =  1  (1 - x2)	arccsc x =  -1  |x| (x2 - 1)
arccos x =   -1  (1 - x2)	arcsec x =  1  |x| (x2 - 1)
arctan x =  1  1 + x2	arccot x =  -1  1 + x2

Hyperbolic

sinh x = cosh x  Proof	csch x = - coth x csch x  Proof
cosh x = sinh x  Proof	sech x = - tanh x sech x  Proof
tanh x = 1 - tanh2 x  Proof	coth  x = 1 - coth2 x  Proof

Referensi bagus:
http://en.wikipedia.org/wiki/Mathematical_joke
http://www.math.ualberta.ca/~runde/jokes.html
http://www.math.utah.edu/~cherk/mathjokes.html
http://www.ahajokes.com

Komik:
http://spikedmath.com/

Salary Theorem
Postulate 1: Knowledge is Power.
Postulate 2: Time is Money.

As every engineer knows:

Power = Work / Time

And since Knowledge = Power and Time = Money
It is therefore true that =

Knowledge = Work / Money .

Solving for Money, we get:

Money = Work / Knowledge.

Thus, as Knowledge approaches zero, Money approaches infinity, regardless of the amount of Work done.

A physicist, a biologist and a mathematician are sitting in a street café watching people entering and leaving the house on the other side of the street. First they see two people entering the house. Time passes. After a while they notice three people leaving the house. The physicist says, "The measurement wasn't accurate." The biologist says, "They must have reproduced." The mathematician says, "If one more person enters the house then it will be empty."