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Equations - General Mathematics

Quadratic roots

f(x) = ax^2 + bx + c

roots = (-b ± sqr(b^2-4ac)) / 2a


Combinations

nCr = n! / ( (n-r)! r! )


Derivatives

sin x cos x

cos x - sin x

tan x sec^2 x


Integrals

x^a x^(a+1) / (a+1), a>1

1/x ln |x|

e^x e^x

a^x a^x / (ln a)

tan x ln |(sec x)|


Simpson's rule

a?b f(x) dx ~=

1/3 . h(y0 + 4y1 + y2)

where, h = 1/2 . (b - a)


Trigonometry/Triangles

sin (A/2) = sqr((s-b)(s-c) / bc)

cos (A/2) = sqr( s(s-a) / bc )

- sine rule

a/(sin A) = b/(sin B) = c/(sin C) = 2R

- cosine rule

a^2 = b^2 + c^2 - 2bc . cos A

- circumcircle

R = abc / 4 . (area of triangle)


Single variable statistics

(number n, mean m) (E sigma, samp x)

derivation, dx = x - m

mean derivation = 1/n . E |dx|

E dx^2 = E x^2 - ((E x)^2) / n

variance = 1/n . E dx^2

standard deviation = sqr(var)


Double variable statistics

correlation coeff =

E (dx dy)/sqr((E dx^2)(E dy^2))


linear regression, y=a+bx

b = E (dx dy) / (E dx^2)

a = (mean y) - b (mean x)


Normal distribution

(mean m, variance s^2)

f(x) = 1/( s . sqr(2 pi) ) .

exp( -(x-m)^2 / (2s^2) )
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