Summary of Zhang Yu's 30 lectures on Advanced Mathematics

1 Preparatory knowledge of Advanced Mathematics

Inverse function

Strictly monotonic functions must have inverse functions , The original function and inverse function are about y=x symmetry .

Four characteristics of functions

1. Boundedness
m <= f(x) <= M
First of all, specify the interval
2. monotonicity

f(x) Is a monotone non decreasing function ----->(x₁-x₂)[f(x₁)-f(x₂)]>=0
f(x) Is a monotone non increasing function ----->(x₁-x₂)[f(x₁)-f(x₂)]<=0

3. Parity

F₁(x)=f(x)-f(-x) Must be an odd function
F₂(x)=f(x)+f(-x) Must be an even function
Function about line x=T symmetrical Sufficient and necessary conditions yes ：
f(x)=f(2T-x) or f(x+T)=f(x-T)

4. periodic

All primitive functions of continuous odd functions are even functions
Among the primitive functions of continuous even functions, only one primitive function is an odd function
f(x) In a finite interval (a,b) It is internally derivable and the derivative function is bounded , Then the original function is bounded in the interval

Function image

arcsinx+arccosx=π/2

r=a(1-cosΘ)


r=asin3Θ

r=aΘ

r²=2a²cos2Θ

Parameter method

Cycloid

x=r(t-sint)
y=r(1-cost)

Star Line

x=rcos³t
y=rsin³t

The sequence

∑k²=(n*(n+1)*(2n+1))/6