Types of mapping?

One-to-one, many-to-one, one-to-many

relationship between f(x) and f-1(x) in terms of domain and range?

f(x) domain = f-1(x) range

f-1(x) domain = f(x) range

Inverse function only exist for ______?

one-to-one function

sec x= _____

cosec x= ______

cot x= _____

1/cos x

1/sin x

1/tan x

Trigonometric identities?

sin^2x+cos^2x=1

1+cot^2x=cosec^2x

1+tan^2x=sec^2x

Trigonometry addition formulae?

sin(A±B)=sinAcosB±cosAsinB

cos(A±B)=cosAcosB-+sinAsinB

Double-angle formula

sin(2A)=2sinAcosA

cos(2A)=cos^2(A)-sin^2(A)=1-2sin^2(A)=2cos^2(A)-1

algorithm of ln?

lna+lnb=ln(ab)

lna-lnb=ln(a/b)

klna=ln(a^k)

ln(e^x)=e^(lnx)=x

Common differentiation:

x^n

sinx

cosx

e^x

a^x

lnx

nx^(n-1)

cosx

-sinx

e^x

a^(x)·lna

1/x

Differentiation Chain rule?

Used in composite function

整体导数✖️内部导数

Differentiation Product rule?

f(x)=uv

f’(x)=uv’+vu’

Differentiation quotient rule?

f(x)=u/v

f’(x)=(vu’-uv’)/v^2

Common Integration:

x^n

sinx

cosx

1/x

a^x

e^x

sec^2x

x^(n+1)/n+1

-cosx

sinx

ln|x|

(a^x)/lna

e^x

tanx

∫f’(ax+b)dx=?

1/af(ax+b)+c

Integration Reverse chain rule?

1) k∫[f(x)^n·f’(x)]dx=?

Try y=[f(x)]^n+1

2) k∫f’(x)/f(x)dx=?

Try y=ln[f(x)]

How to determine if f(x) has one root?

If the function f(x) is continuous on the interval [a, b], f(a) and f(b) have opposite signs, then f(x) has at least one root, x, which satisfies a

Iterative formula?

xn+1=g(xn)