1 derivative
The slope of the curve: Derivative
The second derivative: the speed variation curve slope
Derivative of commonly used functions:
C , = 0 (xn) , = nx n-1 (sinx) , = cosx (cosx) , = -sinx
(ax),=axlna (ex),=ex (logax),=1/xlogae (lnx),=1/x
(u+v),=u,+v, (uv),=u,v+uv,
2 refers to the processing power function routine
f (x) = X X , seeking f (x) Min
t=xx
Taking the logarithm of both sides: lnt = xlnx
Derivative of both sides of x: (. 1 / T) T , = + LNX. 1
Order t, = 0, lnx + 1 = 0
To give: E = X -1
To give E = T -1 / E
2 Taylor formula
f(x)=f(x0)+f,(x0)(x-x0)+f,,(x0)(x-x0)2/2!+…+f(n)(x0)(x-x0)n/n!+Rn(x)
application:
ex=1+x+x2/2!+x3/3!+…xn/n!+Rn
sinx=x-x3/3!+x5/5!-x7/7!+…+(-1)m-1x2m-1/(2m-1)!+R2m
3 gradient
Definition: This function changes the fastest point in the direction
Convex function 4
Definition: If the domain domf function f is a convex set and satisfy x, y belongs domf, 0 <= θ <= 1, there is f (θx + (1-θ) y) <= θf (x) + (1- θ) f (y), the function f is convex.
A first order differentiable: If f first order differentiable, the function when is a convex function and only if the domain domf f a convex set, and x, y belongs domf, f (y)> = f (x) + ▽ f (x) (yx)
Second order differentiable: if the second order differentiable function f, the function f is convex if and only if the function is a convex set dom, and ▽ to 2 f (X)> = 0
5 Probability Theory
PDF: probability density function
Classical probability: n balls into N (N> = n) in the boxes, each box event A = {a probability of at most ball}
N（N-1）…(N-n+1)/(N^^n)
Birthday paradox: 50 students, the same birthday probability that at least 2 people is how much?
1-P50365/365^^50
Packing Problems: 12 and the three defective genuine random packed in three boxes, each containing 5, each box exactly the probability of a defective
3！*12！/4!*4!*4!/(15!/(5!*5!*5!))