关于机器学习的数学基础-高数、线性代数、几率论与数理统计

2019年12月06日 阅读数:740
这篇文章主要向大家介绍关于机器学习的数学基础-高数、线性代数、几率论与数理统计,主要内容包括基础应用、实用技巧、原理机制等方面,希望对大家有所帮助。

高等数学

1.导数定义:html

导数和微分的概念web

f ( x 0 ) = lim Δ x 0   f ( x 0 + Δ x ) f ( x 0 ) Δ x f'({{x}_{0}})=\underset{\Delta x\to 0}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x} (1)算法

或者:app

f ( x 0 ) = lim x x 0   f ( x ) f ( x 0 ) x x 0 f'({{x}_{0}})=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}} (2)ide

2.左右导数导数的几何意义和物理意义svg

函数 f ( x ) f(x) x 0 x_0 处的左、右导数分别定义为:函数

左导数: f ( x 0 ) = lim Δ x 0   f ( x 0 + Δ x ) f ( x 0 ) Δ x = lim x x 0   f ( x ) f ( x 0 ) x x 0 , ( x = x 0 + Δ x ) {{{f}'}_{-}}({{x}_{0}})=\underset{\Delta x\to {{0}^{-}}}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x}=\underset{x\to x_{0}^{-}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}},(x={{x}_{0}}+\Delta x) spa

右导数: f + ( x 0 ) = lim Δ x 0 +   f ( x 0 + Δ x ) f ( x 0 ) Δ x = lim x x 0 +   f ( x ) f ( x 0 ) x x 0 {{{f}'}_{+}}({{x}_{0}})=\underset{\Delta x\to {{0}^{+}}}{\mathop{\lim }}\,\frac{f({{x}_{0}}+\Delta x)-f({{x}_{0}})}{\Delta x}=\underset{x\to x_{0}^{+}}{\mathop{\lim }}\,\frac{f(x)-f({{x}_{0}})}{x-{{x}_{0}}} orm

3.函数的可导性与连续性之间的关系xml

Th1: 函数 f ( x ) f(x) x 0 x_0 处可微 f ( x ) \Leftrightarrow f(x) x 0 x_0 处可导

Th2: 若函数在点 x 0 x_0 处可导,则 y = f ( x ) y=f(x) 在点 x 0 x_0 处连续,反之则不成立。即函数连续不必定可导。

Th3: f ( x 0 ) {f}'({{x}_{0}}) 存在 f ( x 0 ) = f + ( x 0 ) \Leftrightarrow {{{f}'}_{-}}({{x}_{0}})={{{f}'}_{+}}({{x}_{0}})

4.平面曲线的切线和法线

切线方程 : y y 0 = f ( x 0 ) ( x x 0 ) y-{{y}_{0}}=f'({{x}_{0}})(x-{{x}_{0}})
法线方程: y y 0 = 1 f ( x 0 ) ( x x 0 ) , f ( x 0 ) 0 y-{{y}_{0}}=-\frac{1}{f'({{x}_{0}})}(x-{{x}_{0}}),f'({{x}_{0}})\ne 0

5.四则运算法则
设函数 u = u ( x ) v = v ( x ) u=u(x),v=v(x) ]在点 x x 可导则
(1) ( u ± v ) = u ± v (u\pm v{)}'={u}'\pm {v}' d ( u ± v ) = d u ± d v d(u\pm v)=du\pm dv
(2) ( u v ) = u v + v u (uv{)}'=u{v}'+v{u}' d ( u v ) = u d v + v d u d(uv)=udv+vdu
(3) ( u v ) = v u u v v 2 ( v 0 ) (\frac{u}{v}{)}'=\frac{v{u}'-u{v}'}{{{v}^{2}}}(v\ne 0) d ( u v ) = v d u u d v v 2 d(\frac{u}{v})=\frac{vdu-udv}{{{v}^{2}}}

6.基本导数与微分表
(1) y = c y=c (常数) y = 0 {y}'=0 d y = 0 dy=0
(2) y = x α y={{x}^{\alpha }} ($\alpha $为实数) y = α x α 1 {y}'=\alpha {{x}^{\alpha -1}} d y = α x α 1 d x dy=\alpha {{x}^{\alpha -1}}dx
(3) y = a x y={{a}^{x}} y = a x ln a {y}'={{a}^{x}}\ln a d y = a x ln a d x dy={{a}^{x}}\ln adx
特例: ( e x ) = e x ({{{e}}^{x}}{)}'={{{e}}^{x}} d ( e x ) = e x d x d({{{e}}^{x}})={{{e}}^{x}}dx

(4) y = 1 x ln a {y}'=\frac{1}{x\ln a}

d y = 1 x ln a d x dy=\frac{1}{x\ln a}dx
特例: y = ln x y=\ln x ( ln x ) = 1 x (\ln x{)}'=\frac{1}{x} d ( ln x ) = 1 x d x d(\ln x)=\frac{1}{x}dx

(5) y = sin x y=\sin x

y = cos x {y}'=\cos x d ( sin x ) = cos x d x d(\sin x)=\cos xdx

(6) y = cos x y=\cos x

y = sin x {y}'=-\sin x d ( cos x ) = sin x d x d(\cos x)=-\sin xdx

(7) y = tan x y=\tan x

y = 1 cos 2 x = sec 2 x {y}'=\frac{1}{{{\cos }^{2}}x}={{\sec }^{2}}x d ( tan x ) = sec 2 x d x d(\tan x)={{\sec }^{2}}xdx
(8) y = cot x y=\cot x y = 1 sin 2 x = csc 2 x {y}'=-\frac{1}{{{\sin }^{2}}x}=-{{\csc }^{2}}x d ( cot x ) = csc 2 x d x d(\cot x)=-{{\csc }^{2}}xdx
(9) y = sec x y=\sec x y = sec x tan x {y}'=\sec x\tan x

d ( sec x ) = sec x tan x d x d(\sec x)=\sec x\tan xdx
(10) y = csc x y=\csc x y = csc x cot x {y}'=-\csc x\cot x

d ( csc x ) = csc x cot x d x d(\csc x)=-\csc x\cot xdx
(11) y = arcsin x y=\arcsin x

y = 1 1 x 2 {y}'=\frac{1}{\sqrt{1-{{x}^{2}}}}

d ( arcsin x ) = 1 1 x 2 d x d(\arcsin x)=\frac{1}{\sqrt{1-{{x}^{2}}}}dx
(12) y = arccos x y=\arccos x

y = 1 1 x 2 {y}'=-\frac{1}{\sqrt{1-{{x}^{2}}}} d ( arccos x ) = 1 1 x 2 d x d(\arccos x)=-\frac{1}{\sqrt{1-{{x}^{2}}}}dx

(13) y = arctan x y=\arctan x

y = 1 1 + x 2 {y}'=\frac{1}{1+{{x}^{2}}} d ( arctan x ) = 1 1 + x 2 d x d(\arctan x)=\frac{1}{1+{{x}^{2}}}dx

(14) y = arc cot x y=\operatorname{arc}\cot x

y = 1 1 + x 2 {y}'=-\frac{1}{1+{{x}^{2}}}

d ( arc cot x ) = 1 1 + x 2 d x d(\operatorname{arc}\cot x)=-\frac{1}{1+{{x}^{2}}}dx
(15) y = s h x y=shx

y = c h x {y}'=chx d ( s h x ) = c h x d x d(shx)=chxdx

(16) y = c h x y=chx

y = s h x {y}'=shx d ( c h x ) = s h x d x d(chx)=shxdx

7.复合函数,反函数,隐函数以及参数方程所肯定的函数的微分法

(1) 反函数的运算法则: 设 y = f ( x ) y=f(x) 在点 x x 的某邻域内单调连续,在点 x x 处可导且 f ( x ) 0 {f}'(x)\ne 0 ,则其反函数在点 x x 所对应的 y y 处可导,而且有 d y d x = 1 d x d y \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}
(2) 复合函数的运算法则:若 μ = φ ( x ) \mu =\varphi (x) 在点 x x 可导,而 y = f ( μ ) y=f(\mu ) 在对应点 μ \mu ( μ = φ ( x ) \mu =\varphi (x) ) 可导,则复合函数 y = f ( φ ( x ) ) y=f(\varphi (x)) 在点 x x 可导,且 y = f ( μ ) φ ( x ) {y}'={f}'(\mu )\cdot {\varphi }'(x)
(3) 隐函数导数 d y d x \frac{dy}{dx} 的求法通常有三种方法:
1)方程两边对 x x 求导,要记住 y y x x 的函数,则 y y 的函数是 x x 的复合函数.例如 1 y \frac{1}{y} y 2 {{y}^{2}} l n y ln y e y {{{e}}^{y}} 等均是 x x 的复合函数.
x x 求导应按复合函数连锁法则作.
2)公式法.由 F ( x , y ) = 0 F(x,y)=0 d y d x = F x ( x , y ) F y ( x , y ) \frac{dy}{dx}=-\frac{{{{{F}'}}_{x}}(x,y)}{{{{{F}'}}_{y}}(x,y)} ,其中, F x ( x , y ) {{{F}'}_{x}}(x,y)
F y ( x , y ) {{{F}'}_{y}}(x,y) 分别表示 F ( x , y ) F(x,y) x x y y 的偏导数
3)利用微分形式不变性

8.经常使用高阶导数公式

(1) ( a x )   ( n ) = a x ln n a ( a > 0 ) ( e x )   ( n ) = e   x ({{a}^{x}}){{\,}^{(n)}}={{a}^{x}}{{\ln }^{n}}a\quad (a>{0})\quad \quad ({{{e}}^{x}}){{\,}^{(n)}}={e}{{\,}^{x}}
(2) ( sin k x )   ( n ) = k n sin ( k x + n π 2 ) (\sin kx{)}{{\,}^{(n)}}={{k}^{n}}\sin (kx+n\cdot \frac{\pi }{{2}})
(3) ( cos k x )   ( n ) = k n cos ( k x + n π 2 ) (\cos kx{)}{{\,}^{(n)}}={{k}^{n}}\cos (kx+n\cdot \frac{\pi }{{2}})
(4) ( x m )   ( n ) = m ( m 1 ) ( m n + 1 ) x m n ({{x}^{m}}){{\,}^{(n)}}=m(m-1)\cdots (m-n+1){{x}^{m-n}}
(5) ( ln x )   ( n ) = ( 1 ) ( n 1 ) ( n 1 ) ! x n (\ln x){{\,}^{(n)}}={{(-{1})}^{(n-{1})}}\frac{(n-{1})!}{{{x}^{n}}}
(6)莱布尼兹公式:若 u ( x )   , v ( x ) u(x)\,,v(x) n n 阶可导,则
( u v ) ( n ) = i = 0 n c n i u ( i ) v ( n i ) {{(uv)}^{(n)}}=\sum\limits_{i={0}}^{n}{c_{n}^{i}{{u}^{(i)}}{{v}^{(n-i)}}} ,其中 u ( 0 ) = u {{u}^{({0})}}=u v ( 0 ) = v {{v}^{({0})}}=v

9.微分中值定理,泰勒公式

Th1:(费马定理)

若函数 f ( x ) f(x) 知足条件:
(1)函数 f ( x ) f(x) x 0 {{x}_{0}} 的某邻域内有定义,而且在此邻域内恒有
f ( x ) f ( x 0 ) f(x)\le f({{x}_{0}}) f ( x ) f ( x 0 ) f(x)\ge f({{x}_{0}}) ,

(2) f ( x ) f(x) x 0 {{x}_{0}} 处可导,则有 f ( x 0 ) = 0 {f}'({{x}_{0}})=0

Th2:(罗尔定理)

设函数 f ( x ) f(x) 知足条件:
(1)在闭区间 [ a , b ] [a,b] 上连续;

(2)在 ( a , b ) (a,b) 内可导;

(3) f ( a ) = f ( b ) f(a)=f(b)

则在 ( a , b ) (a,b) 内一存在个$\xi $,使 f ( ξ ) = 0 {f}'(\xi )=0
Th3: (拉格朗日中值定理)

设函数 f ( x ) f(x) 知足条件:
(1)在 [ a , b ] [a,b] 上连续;

(2)在 ( a , b ) (a,b) 内可导;

则在 ( a , b ) (a,b) 内一存在个$\xi $,使 f ( b ) f ( a ) b a = f ( ξ ) \frac{f(b)-f(a)}{b-a}={f}'(\xi )

Th4: (柯西中值定理)

设函数 f ( x ) f(x) g ( x ) g(x) 知足条件:
(1) 在 [ a , b ] [a,b] 上连续;

(2) 在 ( a , b ) (a,b) 内可导且 f ( x ) {f}'(x) g ( x ) {g}'(x) 均存在,且 g ( x ) 0 {g}'(x)\ne 0

则在 ( a , b ) (a,b) 内存在一个$\xi $,使 f ( b ) f ( a ) g ( b ) g ( a ) = f ( ξ ) g ( ξ ) \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{{f}'(\xi )}{{g}'(\xi )}

10.洛必达法则
法则Ⅰ ( 0 0 \frac{0}{0} 型)
设函数 f ( x ) , g ( x ) f\left( x \right),g\left( x \right) 知足条件:
lim x x 0   f ( x ) = 0 , lim x x 0   g ( x ) = 0 \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=0 ;

f ( x ) , g ( x ) f\left( x \right),g\left( x \right) x 0 {{x}_{0}} 的邻域内可导,(在 x 0 {{x}_{0}} 处可除外)且 g ( x ) 0 {g}'\left( x \right)\ne 0 ;

lim x x 0   f ( x ) g ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} 存在(或$\infty $)。

则:
lim x x 0   f ( x ) g ( x ) = lim x x 0   f ( x ) g ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}
法则 I {{I}'} ( 0 0 \frac{0}{0} 型)设函数 f ( x ) , g ( x ) f\left( x \right),g\left( x \right) 知足条件:
lim x   f ( x ) = 0 , lim x   g ( x ) = 0 \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to \infty }{\mathop{\lim }}\,g\left( x \right)=0 ;

存在一个 X > 0 X>0 ,当 x > X \left| x \right|>X 时, f ( x ) , g ( x ) f\left( x \right),g\left( x \right) 可导,且 g ( x ) 0 {g}'\left( x \right)\ne 0 ; lim x x 0   f ( x ) g ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} 存在(或 \infty )。

则:
lim x x 0   f ( x ) g ( x ) = lim x x 0   f ( x ) g ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}
法则Ⅱ( \frac{\infty }{\infty } 型) 设函数 f ( x ) , g ( x ) f\left( x \right),g\left( x \right) 知足条件:
lim x x 0   f ( x ) = , lim x x 0   g ( x ) = \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=\infty ,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=\infty ; f ( x ) , g ( x ) f\left( x \right),g\left( x \right) x 0 {{x}_{0}} 的邻域内可导(在 x 0 {{x}_{0}} 处可除外)且 g ( x ) 0 {g}'\left( x \right)\ne 0 ; lim x x 0   f ( x ) g ( x ) \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)} 存在(或 \infty )。则
lim x x 0   f ( x ) g ( x ) = lim x x 0   f ( x ) g ( x ) . \underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}. 同理法则 I I {I{I}'} ( \frac{\infty }{\infty } 型)仿法则 I {{I}'} 可写出。

11.泰勒公式

设函数 f ( x ) f(x) 在点 x 0 {{x}_{0}} 处的某邻域内具备 n + 1 n+1 阶导数,则对该邻域内异于 x 0 {{x}_{0}} 的任意点 x x ,在 x 0 {{x}_{0}} x x 之间至少存在
一个$\xi $,使得:
f ( x ) = f ( x 0 ) + f ( x 0 ) ( x x 0 ) + 1 2 ! f ( x 0 ) ( x x 0 ) 2 + f(x)=f({{x}_{0}})+{f}'({{x}_{0}})(x-{{x}_{0}})+\frac{1}{2!}{f}''({{x}_{0}}){{(x-{{x}_{0}})}^{2}}+\cdots
+ f ( n ) ( x 0 ) n ! ( x x 0 ) n + R n ( x ) +\frac{{{f}^{(n)}}({{x}_{0}})}{n!}{{(x-{{x}_{0}})}^{n}}+{{R}_{n}}(x)
其中 R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x x 0 ) n + 1 {{R}_{n}}(x)=\frac{{{f}^{(n+1)}}(\xi )}{(n+1)!}{{(x-{{x}_{0}})}^{n+1}} 称为 f ( x ) f(x) 在点 x 0 {{x}_{0}} 处的 n n 阶泰勒余项。

x 0 = 0 {{x}_{0}}=0 ,则 n n 阶泰勒公式
f ( x ) = f ( 0 ) + f ( 0 ) x + 1 2 ! f ( 0 ) x 2 + + f ( n ) ( 0 ) n ! x n + R n ( x ) f(x)=f(0)+{f}'(0)x+\frac{1}{2!}{f}''(0){{x}^{2}}+\cdots +\frac{{{f}^{(n)}}(0)}{n!}{{x}^{n}}+{{R}_{n}}(x) ……(1)
其中 R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! x n + 1 {{R}_{n}}(x)=\frac{{{f}^{(n+1)}}(\xi )}{(n+1)!}{{x}^{n+1}} ,$\xi 0 在0与 x$之间.(1)式称为麦克劳林公式

经常使用五种函数在 x 0 = 0 {{x}_{0}}=0 处的泰勒公式

(1) e x = 1 + x + 1 2 ! x 2 + + 1 n ! x n + x n + 1 ( n + 1 ) ! e ξ {{{e}}^{x}}=1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+\frac{{{x}^{n+1}}}{(n+1)!}{{e}^{\xi }}

= 1 + x + 1 2 ! x 2 + + 1 n ! x n + o ( x n ) =1+x+\frac{1}{2!}{{x}^{2}}+\cdots +\frac{1}{n!}{{x}^{n}}+o({{x}^{n}})

(2) sin x = x 1 3 ! x 3 + + x n n ! sin n π 2 + x n + 1 ( n + 1 ) ! sin ( ξ + n + 1 2 π ) \sin x=x-\frac{1}{3!}{{x}^{3}}+\cdots +\frac{{{x}^{n}}}{n!}\sin \frac{n\pi }{2}+\frac{{{x}^{n+1}}}{(n+1)!}\sin (\xi +\frac{n+1}{2}\pi )

= x 1 3 ! x 3 + + x n n ! sin n π 2 + o ( x n ) =x-\frac{1}{3!}{{x}^{3}}+\cdots +\frac{{{x}^{n}}}{n!}\sin \frac{n\pi }{2}+o({{x}^{n}})

(3) cos x = 1 1 2 ! x 2 + + x n n ! cos n π 2 + x n + 1 ( n + 1 ) ! cos ( ξ + n + 1 2 π ) \cos x=1-\frac{1}{2!}{{x}^{2}}+\cdots +\frac{{{x}^{n}}}{n!}\cos \frac{n\pi }{2}+\frac{{{x}^{n+1}}}{(n+1)!}\cos (\xi +\frac{n+1}{2}\pi )

= 1 1 2 ! x 2 + + x n n ! cos n π 2 + o ( x n ) =1-\frac{1}{2!}{{x}^{2}}+\cdots +\frac{{{x}^{n}}}{n!}\cos \frac{n\pi }{2}+o({{x}^{n}})

(4) ln ( 1 + x ) = x 1 2 x 2 + 1 3 x 3 + ( 1 ) n 1 x n n + ( 1 ) n x n + 1 ( n + 1 ) ( 1 + ξ ) n + 1 \ln (1+x)=x-\frac{1}{2}{{x}^{2}}+\frac{1}{3}{{x}^{3}}-\cdots +{{(-1)}^{n-1}}\frac{{{x}^{n}}}{n}+\frac{{{(-1)}^{n}}{{x}^{n+1}}}{(n+1){{(1+\xi )}^{n+1}}}

= x 1 2 x 2 + 1 3 x 3 + ( 1 ) n 1 x n n + o ( x n ) =x-\frac{1}{2}{{x}^{2}}+\frac{1}{3}{{x}^{3}}-\cdots +{{(-1)}^{n-1}}\frac{{{x}^{n}}}{n}+o({{x}^{n}})

(5) ( 1 + x ) m = 1 + m x + m ( m 1 ) 2 ! x 2 + + m ( m 1 ) ( m n + 1 ) n ! x n {{(1+x)}^{m}}=1+mx+\frac{m(m-1)}{2!}{{x}^{2}}+\cdots +\frac{m(m-1)\cdots (m-n+1)}{n!}{{x}^{n}}
+ m ( m 1 ) ( m n + 1 ) ( n + 1 ) ! x n + 1 ( 1 + ξ ) m n 1 +\frac{m(m-1)\cdots (m-n+1)}{(n+1)!}{{x}^{n+1}}{{(1+\xi )}^{m-n-1}}

( 1 + x ) m = 1 + m x + m ( m 1 ) 2 ! x 2 + {{(1+x)}^{m}}=1+mx+\frac{m(m-1)}{2!}{{x}^{2}}+\cdots + m ( m 1 ) ( m n + 1 ) n ! x n + o ( x n ) +\frac{m(m-1)\cdots (m-n+1)}{n!}{{x}^{n}}+o({{x}^{n}})

12.函数单调性的判断
Th1: 设函数 f ( x ) f(x) ( a , b ) (a,b) 区间内可导,若是对 x ( a , b ) \forall x\in (a,b) ,都有 f   ( x ) > 0 f\,'(x)>0 (或 f   ( x ) < 0 f\,'(x)<0 ),则函数 f ( x ) f(x) ( a , b ) (a,b) 内是单调增长的(或单调减小)

Th2: (取极值的必要条件)设函数 f ( x ) f(x) x 0 {{x}_{0}} 处可导,且在 x 0 {{x}_{0}} 处取极值,则 f   ( x 0 ) = 0 f\,'({{x}_{0}})=0

Th3: (取极值的第一充分条件)设函数 f ( x ) f(x) x 0 {{x}_{0}} 的某一邻域内可微,且 f   ( x 0 ) = 0 f\,'({{x}_{0}})=0 (或 f ( x ) f(x) x 0 {{x}_{0}} 处连续,但 f   ( x 0 ) f\,'({{x}_{0}}) 不存在。)
(1)若当 x x 通过 x 0 {{x}_{0}} 时, f   ( x ) f\,'(x) 由“+”变“-”,则 f ( x 0 ) f({{x}_{0}}) 为极大值;
(2)若当 x x​ 通过 x 0 {{x}_{0}}​ 时, f   ( x ) f\,'(x) 由“-”变“+”,则 f ( x 0 ) f({{x}_{0}}) 为极小值;
(3)若 f   ( x ) f\,'(x) 通过 x = x 0 x={{x}_{0}} 的两侧不变号,则 f ( x 0 ) f({{x}_{0}}) 不是极值。

Th4: (取极值的第二充分条件)设 f ( x ) f(x) 在点 x 0 {{x}_{0}} 处有 f ( x ) 0 f''(x)\ne 0 ,且 f   ( x 0 ) = 0 f\,'({{x}_{0}})=0 ,则 当 f   ( x 0 ) < 0 f'\,'({{x}_{0}})<0 时, f ( x 0 ) f({{x}_{0}}) 为极大值;
f   ( x 0 ) > 0 f'\,'({{x}_{0}})>0 时, f ( x 0 ) f({{x}_{0}}) 为极小值。
注:若是 f   ( x 0 ) < 0 f'\,'({{x}_{0}})<0 ,此方法失效。

13.渐近线的求法
(1)水平渐近线 若 lim x +   f ( x ) = b \underset{x\to +\infty }{\mathop{\lim }}\,f(x)=b ,或 lim x   f ( x ) = b \underset{x\to -\infty }{\mathop{\lim }}\,f(x)=b ,则

y = b y=b 称为函数 y = f ( x ) y=f(x) 的水平渐近线。

(2)铅直渐近线 若 lim x x 0   f ( x ) = \underset{x\to x_{0}^{-}}{\mathop{\lim }}\,f(x)=\infty ,或 lim x x 0 +   f ( x ) = \underset{x\to x_{0}^{+}}{\mathop{\lim }}\,f(x)=\infty ,则

x = x 0 x={{x}_{0}} 称为 y = f ( x ) y=f(x) 的铅直渐近线。

(3)斜渐近线 若 a = lim x   f ( x ) x , b = lim x   [ f ( x ) a x ] a=\underset{x\to \infty }{\mathop{\lim }}\,\frac{f(x)}{x},\quad b=\underset{x\to \infty }{\mathop{\lim }}\,[f(x)-ax] ,则
y = a x + b y=ax+b 称为 y = f ( x ) y=f(x) 的斜渐近线。

14.函数凹凸性的判断
Th1: (凹凸性的判别定理)若在I上 f ( x ) < 0 f''(x)<0 (或 f ( x ) > 0 f''(x)>0 ),则 f ( x ) f(x) 在I上是凸的(或凹的)。

Th2: (拐点的判别定理1)若在 x 0 {{x}_{0}} f ( x ) = 0 f''(x)=0 ,(或 f ( x ) f''(x) 不存在),当 x x 变更通过 x 0 {{x}_{0}} 时, f ( x ) f''(x) 变号,则 ( x 0 , f ( x 0 ) ) ({{x}_{0}},f({{x}_{0}})) 为拐点。

Th3: (拐点的判别定理2)设 f ( x ) f(x) x 0 {{x}_{0}} 点的某邻域内有三阶导数,且 f ( x ) = 0 f''(x)=0 f ( x ) 0 f'''(x)\ne 0 ,则 ( x 0 , f ( x 0 ) ) ({{x}_{0}},f({{x}_{0}})) 为拐点。

15.弧微分

d S = 1 + y 2 d x dS=\sqrt{1+y{{'}^{2}}}dx

线性代数

行列式

1.行列式按行(列)展开定理

(1) 设 A = ( a i j ) n × n A = ( a_{{ij}} )_{n \times n} ,则: a i 1 A j 1 + a i 2 A j 2 + + a i n A j n = { A , i = j 0 , i j a_{i1}A_{j1} +a_{i2}A_{j2} + \cdots + a_{{in}}A_{{jn}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases}

a 1 i A 1 j + a 2 i A 2 j + + a n i A n j = { A , i = j 0 , i j a_{1i}A_{1j} + a_{2i}A_{2j} + \cdots + a_{{ni}}A_{{nj}} = \begin{cases}|A|,i=j\\ 0,i \neq j\end{cases} A A = A A = A E , AA^{*} = A^{*}A = \left| A \right|E, 其中: A = ( A 11 A 12 A 1 n A 21 A 22 A 2 n A n 1 A n 2 A n n ) = ( A j i ) = ( A i j ) T A^{*} = \begin{pmatrix} A_{11} & A_{12} & \ldots & A_{1n} \\ A_{21} & A_{22} & \ldots & A_{2n} \\ \ldots & \ldots & \ldots & \ldots \\ A_{n1} & A_{n2} & \ldots & A_{{nn}} \\ \end{pmatrix} = (A_{{ji}}) = {(A_{{ij}})}^{T}

D n = 1 1 1 x 1 x 2 x n x 1 n 1 x 2 n 1 x n n 1 = 1 j < i n   ( x i x j ) D_{n} = \begin{vmatrix} 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & \ldots & x_{n} \\ \ldots & \ldots & \ldots & \ldots \\ x_{1}^{n - 1} & x_{2}^{n - 1} & \ldots & x_{n}^{n - 1} \\ \end{vmatrix} = \prod_{1 \leq j < i \leq n}^{}\,(x_{i} - x_{j})

(2) 设 A , B A,B n n 阶方阵,则 A B = A B = B A = B A \left| {AB} \right| = \left| A \right|\left| B \right| = \left| B \right|\left| A \right| = \left| {BA} \right| ,但 A ± B = A ± B \left| A \pm B \right| = \left| A \right| \pm \left| B \right| 不必定成立。

(3) k A = k n A \left| {kA} \right| = k^{n}\left| A \right| , A A n n 阶方阵。

(4) 设 A A n n 阶方阵, A T = A ; A 1 = A 1 |A^{T}| = |A|;|A^{- 1}| = |A|^{- 1} (若 A A 可逆), A = A n 1 |A^{*}| = |A|^{n - 1}

n 2 n \geq 2

(5) A O O B = A C O B = A O C B = A B \left| \begin{matrix} & {A\quad O} \\ & {O\quad B} \\ \end{matrix} \right| = \left| \begin{matrix} & {A\quad C} \\ & {O\quad B} \\ \end{matrix} \right| = \left| \begin{matrix} & {A\quad O} \\ & {C\quad B} \\ \end{matrix} \right| =| A||B|
A , B A,B 为方阵,但 O A m × m B n × n O = ( 1 ) m n A B \left| \begin{matrix} {O} & A_{m \times m} \\ B_{n \times n} & { O} \\ \end{matrix} \right| = ({- 1)}^{{mn}}|A||B|

(6) 范德蒙行列式 D n = 1 1 1 x 1 x 2 x n x 1 n 1 x 2 n 1 x n n 1 = 1 j < i n   ( x i x j ) D_{n} = \begin{vmatrix} 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & \ldots & x_{n} \\ \ldots & \ldots & \ldots & \ldots \\ x_{1}^{n - 1} & x_{2}^{n 1} & \ldots & x_{n}^{n - 1} \\ \end{vmatrix} = \prod_{1 \leq j < i \leq n}^{}\,(x_{i} - x_{j})

A A n n 阶方阵, λ i ( i = 1 , 2   , n ) \lambda_{i}(i = 1,2\cdots,n) A A n n 个特征值,则
A = i = 1 n λ i |A| = \prod_{i = 1}^{n}\lambda_{i}​

矩阵

矩阵: m × n m \times n 个数 a i j a_{{ij}} 排成 m m n n 列的表格 [ a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ] \begin{bmatrix} a_{11}\quad a_{12}\quad\cdots\quad a_{1n} \\ a_{21}\quad a_{22}\quad\cdots\quad a_{2n} \\ \quad\cdots\cdots\cdots\cdots\cdots \\ a_{m1}\quad a_{m2}\quad\cdots\quad a_{{mn}} \\ \end{bmatrix} 称为矩阵,简记为 A A ,或者 ( a i j ) m × n \left( a_{{ij}} \right)_{m \times n} 。若 m = n m = n ,则称 A A n n 阶矩阵或 n n 阶方阵。

矩阵的线性运算

1.矩阵的加法

A = ( a i j ) , B = ( b i j ) A = (a_{{ij}}),B = (b_{{ij}}) 是两个 m × n m \times n 矩阵,则 m × n m \times n 矩阵 C = c i j ) = a i j + b i j C = c_{{ij}}) = a_{{ij}} + b_{{ij}} 称为矩阵 A A B B 的和,记为 A + B = C A + B = C

2.矩阵的数乘

A = ( a i j ) A = (a_{{ij}}) m × n m \times n 矩阵, k k 是一个常数,则 m × n m \times n 矩阵 ( k a i j ) (ka_{{ij}}) 称为数 k k 与矩阵 A A 的数乘,记为 k A {kA}

3.矩阵的乘法

A = ( a i j ) A = (a_{{ij}}) m × n m \times n 矩阵, B = ( b i j ) B = (b_{{ij}}) n × s n \times s 矩阵,那么 m × s m \times s 矩阵 C = ( c i j ) C = (c_{{ij}}) ,其中 c i j = a i 1 b 1 j + a i 2 b 2 j + + a i n b n j = k = 1 n a i k b k j c_{{ij}} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{{in}}b_{{nj}} = \sum_{k =1}^{n}{a_{{ik}}b_{{kj}}} 称为 A B {AB} 的乘积,记为 C = A B C = AB

4. A T \mathbf{A}^{\mathbf{T}} A 1 \mathbf{A}^{\mathbf{-1}} A \mathbf{A}^{\mathbf{*}} 三者之间的关系

(1) ( A T ) T = A , ( A B ) T = B T A T , ( k A ) T = k A T , ( A ± B ) T = A T ± B T {(A^{T})}^{T} = A,{(AB)}^{T} = B^{T}A^{T},{(kA)}^{T} = kA^{T},{(A \pm B)}^{T} = A^{T} \pm B^{T}

(2) ( A 1 ) 1 = A , ( A B ) 1 = B 1 A 1 , ( k A ) 1 = 1 k A 1 , \left( A^{- 1} \right)^{- 1} = A,\left( {AB} \right)^{- 1} = B^{- 1}A^{- 1},\left( {kA} \right)^{- 1} = \frac{1}{k}A^{- 1},

( A ± B ) 1 = A 1 ± B 1 {(A \pm B)}^{- 1} = A^{- 1} \pm B^{- 1} 不必定成立。

(3) ( A ) = A n 2   A    ( n 3 ) \left( A^{*} \right)^{*} = |A|^{n - 2}\ A\ \ (n \geq 3) ( A B ) = B A , \left({AB} \right)^{*} = B^{*}A^{*}, ( k A ) = k n 1 A    ( n 2 ) \left( {kA} \right)^{*} = k^{n -1}A^{*}{\ \ }\left( n \geq 2 \right)

( A ± B ) = A ± B \left( A \pm B \right)^{*} = A^{*} \pm B^{*} 不必定成立。

(4) ( A 1 ) T = ( A T ) 1 ,   ( A 1 ) = ( A A ) 1 , ( A ) T = ( A T ) {(A^{- 1})}^{T} = {(A^{T})}^{- 1},\ \left( A^{- 1} \right)^{*} ={(AA^{*})}^{- 1},{(A^{*})}^{T} = \left( A^{T} \right)^{*}

5.有关 A \mathbf{A}^{\mathbf{*}} 的结论

(1) A A = A A = A E AA^{*} = A^{*}A = |A|E

(2) A = A n 1   ( n 2 ) ,      ( k A ) = k n 1 A ,    ( A ) = A n 2 A ( n 3 ) |A^{*}| = |A|^{n - 1}\ (n \geq 2),\ \ \ \ {(kA)}^{*} = k^{n -1}A^{*},{{\ \ }\left( A^{*} \right)}^{*} = |A|^{n - 2}A(n \geq 3)

(3) 若 A A 可逆,则 A = A A 1 , ( A ) = 1 A A A^{*} = |A|A^{- 1},{(A^{*})}^{*} = \frac{1}{|A|}A

(4) 若 A A​ n n​ 阶方阵,则:

r ( A ) = { n , r ( A ) = n 1 , r ( A ) = n 1 0 , r ( A ) < n 1 r(A^*)=\begin{cases}n,\quad r(A)=n\\ 1,\quad r(A)=n-1\\ 0,\quad r(A)<n-1\end{cases}

6.有关 A 1 \mathbf{A}^{\mathbf{- 1}} 的结论

A A 可逆 A B = E ; A 0 ; r ( A ) = n ; \Leftrightarrow AB = E; \Leftrightarrow |A| \neq 0; \Leftrightarrow r(A) = n;

A \Leftrightarrow A 能够表示为初等矩阵的乘积; A ; A x = 0 \Leftrightarrow A;\Leftrightarrow Ax = 0

7.有关矩阵秩的结论

(1) 秩 r ( A ) r(A) =行秩=列秩;

(2) r ( A m × n ) min ( m , n ) ; r(A_{m \times n}) \leq \min(m,n);

(3) A 0 r ( A ) 1 A \neq 0 \Rightarrow r(A) \geq 1

(4) r ( A ± B ) r ( A ) + r ( B ) ; r(A \pm B) \leq r(A) + r(B);

(5) 初等变换不改变矩阵的秩

(6) r ( A ) + r ( B ) n r ( A B ) min ( r ( A ) , r ( B ) ) , r(A) + r(B) - n \leq r(AB) \leq \min(r(A),r(B)), 特别若 A B = O AB = O
则: r ( A ) + r ( B ) n r(A) + r(B) \leq n

(7) 若 A 1 A^{- 1} 存在 r ( A B ) = r ( B ) ; \Rightarrow r(AB) = r(B); B 1 B^{- 1} 存在
r ( A B ) = r ( A ) ; \Rightarrow r(AB) = r(A);

r ( A m × n ) = n r ( A B ) = r ( B ) ; r(A_{m \times n}) = n \Rightarrow r(AB) = r(B); r ( A m × s ) = n r ( A B ) = r ( A ) r(A_{m \times s}) = n\Rightarrow r(AB) = r\left( A \right)

(8) r ( A m × s ) = n A x = 0 r(A_{m \times s}) = n \Leftrightarrow Ax = 0 只有零解

8.分块求逆公式

( A O O B ) 1 = ( A 1 O O B 1 ) \begin{pmatrix} A & O \\ O & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} A^{-1} & O \\ O & B^{- 1} \\ \end{pmatrix} ( A C O B ) 1 = ( A