数学基础知识
数据科学需要一定的数学基础,但仅仅做应用的话,如果时间不多,不用学太深,了解基本公式即可,遇到问题再查吧。
下面是常见的一些数学基础概念,建议大家收藏后再仔细阅读,遇到不懂的概念可以直接在这里查~
高等数学
1.导数定义:
导数和微分的概念
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}
f ′ ( x 0 ) = Δ x → 0 lim Δ x f ( x 0 + Δ x ) − f ( x 0 ) (1)
或者:
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}}}
f ′ ( x 0 ) = x → x 0 lim x − x 0 f ( x ) − f ( x 0 ) (2)
2.左右导数导数的几何意义和物理意义
函数
f
(
x
)
f(x)
f ( x ) 在
x
0
x_0
x 0 处的左、右导数分别定义为:
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)
f ′ − ( x 0 ) = Δ x → 0 − lim Δ x f ( x 0 + Δ x ) − f ( x 0 ) = x → x 0 − lim x − x 0 f ( x ) − f ( x 0 ) , ( x = x 0 + Δ x )
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}}}
f ′ + ( x 0 ) = Δ x → 0 + lim Δ x f ( x 0 + Δ x ) − f ( x 0 ) = x → x 0 + lim x − x 0 f ( x ) − f ( x 0 )
3.函数的可导性与连续性之间的关系
Th1: 函数
f
(
x
)
f(x)
f ( x ) 在
x
0
x_0
x 0 处可微
⇔
f
(
x
)
\Leftrightarrow f(x)
⇔ f ( x ) 在
x
0
x_0
x 0 处可导
Th2: 若函数在点
x
0
x_0
x 0 处可导,则
y
=
f
(
x
)
y=f(x)
y = f ( x ) 在点
x
0
x_0
x 0 处连续,反之则不成立。即函数连续不一定可导。
Th3:
f
′
(
x
0
)
{f}'({{x}_{0}})
f ′ ( x 0 ) 存在
⇔
f
′
−
(
x
0
)
=
f
′
+
(
x
0
)
\Leftrightarrow {{{f}'}_{-}}({{x}_{0}})={{{f}'}_{+}}({{x}_{0}})
⇔ 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}})
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
y − y 0 = − f ′ ( x 0 ) 1 ( x − x 0 ) , f ′ ( x 0 ) = 0
5.四则运算法则 设函数
u
=
u
(
x
)
,
v
=
v
(
x
)
u=u(x),v=v(x)
u = u ( x ) , v = v ( x ) ]在点
x
x
x 可导则 (1)
(
u
±
v
)
′
=
u
′
±
v
′
(u\pm v{)}'={u}'\pm {v}'
( u ± v ) ′ = u ′ ± v ′
d
(
u
±
v
)
=
d
u
±
d
v
d(u\pm v)=du\pm dv
d ( u ± v ) = d u ± d v (2)
(
u
v
)
′
=
u
v
′
+
v
u
′
(uv{)}'=u{v}'+v{u}'
( u v ) ′ = u v ′ + v u ′
d
(
u
v
)
=
u
d
v
+
v
d
u
d(uv)=udv+vdu
d ( u v ) = u d v + v d u (3)
(
u
v
)
′
=
v
u
′
−
u
v
′
v
2
(
v
≠
0
)
(\frac{u}{v}{)}'=\frac{v{u}'-u{v}'}{{{v}^{2}}}(v\ne 0)
( v u ) ′ = v 2 v u ′ − u v ′ ( v = 0 )
d
(
u
v
)
=
v
d
u
−
u
d
v
v
2
d(\frac{u}{v})=\frac{vdu-udv}{{{v}^{2}}}
d ( v u ) = v 2 v d u − u d v
6.基本导数与微分表 (1)
y
=
c
y=c
y = c (常数)
y
′
=
0
{y}'=0
y ′ = 0
d
y
=
0
dy=0
d y = 0 (2)
y
=
x
α
y={{x}^{\alpha }}
y = x α (
α
\alpha
α 为实数)
y
′
=
α
x
α
−
1
{y}'=\alpha {{x}^{\alpha -1}}
y ′ = α x α − 1
d
y
=
α
x
α
−
1
d
x
dy=\alpha {{x}^{\alpha -1}}dx
d y = α x α − 1 d x (3)
y
=
a
x
y={{a}^{x}}
y = a x
y
′
=
a
x
ln
a
{y}'={{a}^{x}}\ln a
y ′ = a x ln a
d
y
=
a
x
ln
a
d
x
dy={{a}^{x}}\ln adx
d y = a x ln a d x 特例:
(
e
x
)
′
=
e
x
({{{e}}^{x}}{)}'={{{e}}^{x}}
( e x ) ′ = e x
d
(
e
x
)
=
e
x
d
x
d({{{e}}^{x}})={{{e}}^{x}}dx
d ( e x ) = e x d x
(4)
y
=
log
a
x
y={{\log }_{a}}x
y = log a x
y
′
=
1
x
ln
a
{y}'=\frac{1}{x\ln a}
y ′ = x ln a 1
d
y
=
1
x
ln
a
d
x
dy=\frac{1}{x\ln a}dx
d y = x ln a 1 d x 特例:
y
=
ln
x
y=\ln x
y = ln x
(
ln
x
)
′
=
1
x
(\ln x{)}'=\frac{1}{x}
( ln x ) ′ = x 1
d
(
ln
x
)
=
1
x
d
x
d(\ln x)=\frac{1}{x}dx
d ( ln x ) = x 1 d x
(5)
y
=
sin
x
y=\sin x
y = sin x
y
′
=
cos
x
{y}'=\cos x
y ′ = cos x
d
(
sin
x
)
=
cos
x
d
x
d(\sin x)=\cos xdx
d ( sin x ) = cos x d x
(6)
y
=
cos
x
y=\cos x
y = cos x
y
′
=
−
sin
x
{y}'=-\sin x
y ′ = − sin x
d
(
cos
x
)
=
−
sin
x
d
x
d(\cos x)=-\sin xdx
d ( cos x ) = − sin x d x
(7)
y
=
tan
x
y=\tan x
y = tan x
y
′
=
1
cos
2
x
=
sec
2
x
{y}'=\frac{1}{{{\cos }^{2}}x}={{\sec }^{2}}x
y ′ = cos 2 x 1 = sec 2 x
d
(
tan
x
)
=
sec
2
x
d
x
d(\tan x)={{\sec }^{2}}xdx
d ( tan x ) = sec 2 x d x (8)
y
=
cot
x
y=\cot x
y = cot x
y
′
=
−
1
sin
2
x
=
−
csc
2
x
{y}'=-\frac{1}{{{\sin }^{2}}x}=-{{\csc }^{2}}x
y ′ = − sin 2 x 1 = − csc 2 x
d
(
cot
x
)
=
−
csc
2
x
d
x
d(\cot x)=-{{\csc }^{2}}xdx
d ( cot x ) = − csc 2 x d x (9)
y
=
sec
x
y=\sec x
y = sec x
y
′
=
sec
x
tan
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
d ( sec x ) = sec x tan x d x (10)
y
=
csc
x
y=\csc x
y = csc x
y
′
=
−
csc
x
cot
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
d ( csc x ) = − csc x cot x d x (11)
y
=
arcsin
x
y=\arcsin x
y = arcsin x
y
′
=
1
1
−
x
2
{y}'=\frac{1}{\sqrt{1-{{x}^{2}}}}
y ′ = 1 − x 2
1
d
(
arcsin
x
)
=
1
1
−
x
2
d
x
d(\arcsin x)=\frac{1}{\sqrt{1-{{x}^{2}}}}dx
d ( arcsin x ) = 1 − x 2
1 d x (12)
y
=
arccos
x
y=\arccos x
y = arccos x
y
′
=
−
1
1
−
x
2
{y}'=-\frac{1}{\sqrt{1-{{x}^{2}}}}
y ′ = − 1 − x 2
1
d
(
arccos
x
)
=
−
1
1
−
x
2
d
x
d(\arccos x)=-\frac{1}{\sqrt{1-{{x}^{2}}}}dx
d ( arccos x ) = − 1 − x 2
1 d x
(13)
y
=
arctan
x
y=\arctan x
y = arctan x
y
′
=
1
1
+
x
2
{y}'=\frac{1}{1+{{x}^{2}}}
y ′ = 1 + x 2 1
d
(
arctan
x
)
=
1
1
+
x
2
d
x
d(\arctan x)=\frac{1}{1+{{x}^{2}}}dx
d ( arctan x ) = 1 + x 2 1 d x
(14)
y
=
arc
cot
x
y=\operatorname{arc}\cot x
y = a r c cot x
y
′
=
−
1
1
+
x
2
{y}'=-\frac{1}{1+{{x}^{2}}}
y ′ = − 1 + x 2 1
d
(
arc
cot
x
)
=
−
1
1
+
x
2
d
x
d(\operatorname{arc}\cot x)=-\frac{1}{1+{{x}^{2}}}dx
d ( a r c cot x ) = − 1 + x 2 1 d x (15)
y
=
s
h
x
y=shx
y = s h x
y
′
=
c
h
x
{y}'=chx
y ′ = c h x
d
(
s
h
x
)
=
c
h
x
d
x
d(shx)=chxdx
d ( s h x ) = c h x d x
(16)
y
=
c
h
x
y=chx
y = c h x
y
′
=
s
h
x
{y}'=shx
y ′ = s h x
d
(
c
h
x
)
=
s
h
x
d
x
d(chx)=shxdx
d ( c h x ) = s h x d x
7.复合函数,反函数,隐函数以及参数方程所确定的函数的微分法
(1) 反函数的运算法则: 设
y
=
f
(
x
)
y=f(x)
y = f ( x ) 在点
x
x
x 的某邻域内单调连续,在点
x
x
x 处可导且
f
′
(
x
)
≠
0
{f}'(x)\ne 0
f ′ ( x ) = 0 ,则其反函数在点
x
x
x 所对应的
y
y
y 处可导,并且有
d
y
d
x
=
1
d
x
d
y
\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}
d x d y = d y d x 1 (2) 复合函数的运算法则:若
μ
=
φ
(
x
)
\mu =\varphi(x)
μ = φ ( x ) 在点
x
x
x 可导,而
y
=
f
(
μ
)
y=f(\mu)
y = f ( μ ) 在对应点
μ
\mu
μ (
μ
=
φ
(
x
)
\mu =\varphi (x)
μ = φ ( x ) )可导,则复合函数
y
=
f
(
φ
(
x
)
)
y=f(\varphi (x))
y = f ( φ ( x ) ) 在点
x
x
x 可导,且
y
′
=
f
′
(
μ
)
⋅
φ
′
(
x
)
{y}'={f}'(\mu )\cdot {\varphi }'(x)
y ′ = f ′ ( μ ) ⋅ φ ′ ( x ) (3) 隐函数导数
d
y
d
x
\frac{dy}{dx}
d x d y 的求法一般有三种方法: 1)方程两边对
x
x
x 求导,要记住
y
y
y 是
x
x
x 的函数,则
y
y
y 的函数是
x
x
x 的复合函数.例如
1
y
\frac{1}{y}
y 1 ,
y
2
{{y}^{2}}
y 2 ,
l
n
y
ln y
l n y ,
e
y
{{{e}}^{y}}
e y 等均是
x
x
x 的复合函数. 对
x
x
x 求导应按复合函数连锁法则做. 2)公式法.由
F
(
x
,
y
)
=
0
F(x,y)=0
F ( x , y ) = 0 知
d
y
d
x
=
−
F
′
x
(
x
,
y
)
F
′
y
(
x
,
y
)
\frac{dy}{dx}=-\frac{{{{{F}'}}_{x}}(x,y)}{{{{{F}'}}_{y}}(x,y)}
d x d y = − F ′ y ( x , y ) F ′ x ( x , y ) ,其中,
F
′
x
(
x
,
y
)
{{{F}'}_{x}}(x,y)
F ′ x ( x , y ) ,
F
′
y
(
x
,
y
)
{{{F}'}_{y}}(x,y)
F ′ y ( x , y ) 分别表示
F
(
x
,
y
)
F(x,y)
F ( x , y ) 对
x
x
x 和
y
y
y 的偏导数 3)利用微分形式不变性
8.常用高阶导数公式
(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}}
( a x ) ( n ) = a x ln n a ( a > 0 ) ( 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}})
( sin k x ) ( n ) = k n sin ( k x + n ⋅ 2 π ) (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}})
( cos k x ) ( n ) = k n cos ( k x + n ⋅ 2 π ) (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}}
( x m ) ( n ) = m ( m − 1 ) ⋯ ( 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}}}
( ln x ) ( n ) = ( − 1 ) ( n − 1 ) x n ( n − 1 ) ! (6)莱布尼兹公式:若
u
(
x
)
,
v
(
x
)
u(x)\,,v(x)
u ( x ) , v ( x ) 均
n
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)}}}
( u v ) ( n ) = i = 0 ∑ n c n i u ( i ) v ( n − i ) ,其中
u
(
0
)
=
u
{{u}^{({0})}}=u
u ( 0 ) = u ,
v
(
0
)
=
v
{{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)
f ( x ) 满足条件: (1)函数
f
(
x
)
f(x)
f ( x ) 在
x
0
{{x}_{0}}
x 0 的某邻域内有定义,并且在此邻域内恒有
f
(
x
)
≤
f
(
x
0
)
f(x)\le f({{x}_{0}})
f ( x ) ≤ f ( x 0 ) 或
f
(
x
)
≥
f
(
x
0
)
f(x)\ge f({{x}_{0}})
f ( x ) ≥ f ( x 0 ) ,
(2)
f
(
x
)
f(x)
f ( x ) 在
x
0
{{x}_{0}}
x 0 处可导,则有
f
′
(
x
0
)
=
0
{f}'({{x}_{0}})=0
f ′ ( x 0 ) = 0
Th2: (罗尔定理)
设函数
f
(
x
)
f(x)
f ( x ) 满足条件: (1)在闭区间
[
a
,
b
]
[a,b]
[ a , b ] 上连续;
(2)在
(
a
,
b
)
(a,b)
( a , b ) 内可导;
(3)
f
(
a
)
=
f
(
b
)
f(a)=f(b)
f ( a ) = f ( b ) ;
则在
(
a
,
b
)
(a,b)
( a , b ) 内一存在个$\xi $,使
f
′
(
ξ
)
=
0
{f}'(\xi )=0
f ′ ( ξ ) = 0 Th3: (拉格朗日中值定理)
设函数
f
(
x
)
f(x)
f ( x ) 满足条件: (1)在
[
a
,
b
]
[a,b]
[ a , b ] 上连续;
(2)在
(
a
,
b
)
(a,b)
( a , b ) 内可导;
则在
(
a
,
b
)
(a,b)
( a , b ) 内一存在个$\xi $,使
f
(
b
)
−
f
(
a
)
b
−
a
=
f
′
(
ξ
)
\frac{f(b)-f(a)}{b-a}={f}'(\xi )
b − a f ( b ) − f ( a ) = f ′ ( ξ )
Th4: (柯西中值定理)
设函数
f
(
x
)
f(x)
f ( x ) ,
g
(
x
)
g(x)
g ( x ) 满足条件: (1) 在
[
a
,
b
]
[a,b]
[ a , b ] 上连续;
(2) 在
(
a
,
b
)
(a,b)
( a , b ) 内可导且
f
′
(
x
)
{f}'(x)
f ′ ( x ) ,
g
′
(
x
)
{g}'(x)
g ′ ( x ) 均存在,且
g
′
(
x
)
≠
0
{g}'(x)\ne 0
g ′ ( x ) = 0
则在
(
a
,
b
)
(a,b)
( a , b ) 内存在一个$\xi $,使
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
=
f
′
(
ξ
)
g
′
(
ξ
)
\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{{f}'(\xi )}{{g}'(\xi )}
g ( b ) − g ( a ) f ( b ) − f ( a ) = g ′ ( ξ ) f ′ ( ξ )
10.洛必达法则 法则Ⅰ (
0
0
\frac{0}{0}
0 0 型) 设函数
f
(
x
)
,
g
(
x
)
f\left( x \right),g\left( x \right)
f ( x ) , g ( x ) 满足条件:
lim
x
→
x
0
f
(
x
)
=
0
,
lim
x
→
x
0
g
(
x
)
=
0
\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=0
x → x 0 lim f ( x ) = 0 , x → x 0 lim g ( x ) = 0 ;
f
(
x
)
,
g
(
x
)
f\left( x \right),g\left( x \right)
f ( x ) , g ( x ) 在
x
0
{{x}_{0}}
x 0 的邻域内可导,(在
x
0
{{x}_{0}}
x 0 处可除外)且
g
′
(
x
)
≠
0
{g}'\left( x \right)\ne 0
g ′ ( x ) = 0 ;
lim
x
→
x
0
f
′
(
x
)
g
′
(
x
)
\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}
x → x 0 lim g ′ ( x ) f ′ ( x ) 存在(或$\infty $)。
则:
lim
x
→
x
0
f
(
x
)
g
(
x
)
=
lim
x
→
x
0
f
′
(
x
)
g
′
(
x
)
\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}
x → x 0 lim g ( x ) f ( x ) = x → x 0 lim g ′ ( x ) f ′ ( x ) 。 法则
I
′
{{I}'}
I ′ (
0
0
\frac{0}{0}
0 0 型)设函数
f
(
x
)
,
g
(
x
)
f\left( x \right),g\left( x \right)
f ( x ) , g ( x ) 满足条件:
lim
x
→
∞
f
(
x
)
=
0
,
lim
x
→
∞
g
(
x
)
=
0
\underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=0,\underset{x\to \infty }{\mathop{\lim }}\,g\left( x \right)=0
x → ∞ lim f ( x ) = 0 , x → ∞ lim g ( x ) = 0 ;
存在一个
X
>
0
X>0
X > 0 ,当
∣
x
∣
>
X
\left| x \right|>X
∣ x ∣ > X 时,
f
(
x
)
,
g
(
x
)
f\left( x \right),g\left( x \right)
f ( x ) , g ( x ) 可导,且
g
′
(
x
)
≠
0
{g}'\left( x \right)\ne 0
g ′ ( x ) = 0 ;
lim
x
→
x
0
f
′
(
x
)
g
′
(
x
)
\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}
x → x 0 lim g ′ ( x ) f ′ ( x ) 存在(或$\infty $)。
则
lim
x
→
x
0
f
(
x
)
g
(
x
)
=
lim
x
→
x
0
f
′
(
x
)
g
′
(
x
)
\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}
x → x 0 lim g ( x ) f ( x ) = x → x 0 lim g ′ ( x ) f ′ ( x ) 法则Ⅱ(
∞
∞
\frac{\infty }{\infty }
∞ ∞ 型) 设函数
f
(
x
)
,
g
(
x
)
f\left( x \right),g\left( x \right)
f ( x ) , g ( x ) 满足条件:
lim
x
→
x
0
f
(
x
)
=
∞
,
lim
x
→
x
0
g
(
x
)
=
∞
\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,f\left( x \right)=\infty,\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,g\left( x \right)=\infty
x → x 0 lim f ( x ) = ∞ , x → x 0 lim g ( x ) = ∞ ;
f
(
x
)
,
g
(
x
)
f\left( x \right),g\left( x \right)
f ( x ) , g ( x ) 在
x
0
{{x}_{0}}
x 0 的邻域内可导(在
x
0
{{x}_{0}}
x 0 处可除外)且
g
′
(
x
)
≠
0
{g}'\left( x \right)\ne 0
g ′ ( x ) = 0 ;
lim
x
→
x
0
f
′
(
x
)
g
′
(
x
)
\underset{x\to {{x}_{0}}}{\mathop{\lim }}\,\frac{{f}'\left( x \right)}{{g}'\left( x \right)}
x → x 0 lim g ′ ( x ) f ′ ( x ) 存在(或$\infty
)
。
则
)。 则
) 。 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)}
x → x 0 lim g ( x ) f ( x ) = x → x 0 lim g ′ ( x ) f ′ ( x ) $ 同理法则
I
I
′
{I{I}'}
I I ′ (
∞
∞
\frac{\infty }{\infty }
∞ ∞ 型)仿法则
I
′
{{I}'}
I ′ 可写出。
11.泰勒公式
设函数
f
(
x
)
f(x)
f ( x ) 在点
x
0
{{x}_{0}}
x 0 处的某邻域内具有
n
+
1
n+1
n + 1 阶导数,则对该邻域内异于
x
0
{{x}_{0}}
x 0 的任意点
x
x
x ,在
x
0
{{x}_{0}}
x 0 与
x
x
x 之间至少存在 一个
ξ
\xi
ξ ,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 ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 2 ! 1 f ′ ′ ( x 0 ) ( x − x 0 ) 2 + ⋯
+
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)
+ n ! f ( n ) ( x 0 ) ( 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}}
R n ( x ) = ( n + 1 ) ! f ( n + 1 ) ( ξ ) ( x − x 0 ) n + 1 称为
f
(
x
)
f(x)
f ( x ) 在点
x
0
{{x}_{0}}
x 0 处的
n
n
n 阶泰勒余项。
令
x
0
=
0
{{x}_{0}}=0
x 0 = 0 ,则
n
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)
f ( x ) = f ( 0 ) + f ′ ( 0 ) x + 2 ! 1 f ′ ′ ( 0 ) x 2 + ⋯ + n ! f ( n ) ( 0 ) 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}}
R n ( x ) = ( n + 1 ) ! f ( n + 1 ) ( ξ ) x n + 1 ,$\xi
在
0
与
在0与
在 0 与 x$之间.(1)式称为麦克劳林公式
常用五种函数在
x
0
=
0
{{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 }}
e x = 1 + x + 2 ! 1 x 2 + ⋯ + n ! 1 x n + ( n + 1 ) ! x n + 1 e ξ
或
=
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}})
= 1 + x + 2 ! 1 x 2 + ⋯ + n ! 1 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 )
sin x = x − 3 ! 1 x 3 + ⋯ + n ! x n sin 2 n π + ( n + 1 ) ! x n + 1 sin ( ξ + 2 n + 1 π )
或
=
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}})
= x − 3 ! 1 x 3 + ⋯ + n ! x n sin 2 n π + 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 )
cos x = 1 − 2 ! 1 x 2 + ⋯ + n ! x n cos 2 n π + ( n + 1 ) ! x n + 1 cos ( ξ + 2 n + 1 π )
或
=
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}})
= 1 − 2 ! 1 x 2 + ⋯ + n ! x n cos 2 n π + 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}}}
ln ( 1 + x ) = x − 2 1 x 2 + 3 1 x 3 − ⋯ + ( − 1 ) n − 1 n x n + ( n + 1 ) ( 1 + ξ ) n + 1 ( − 1 ) n x n + 1
或
=
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}})
= x − 2 1 x 2 + 3 1 x 3 − ⋯ + ( − 1 ) n − 1 n x 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}}
( 1 + x ) m = 1 + m x + 2 ! m ( m − 1 ) x 2 + ⋯ + n ! m ( m − 1 ) ⋯ ( m − n + 1 ) 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}}
+ ( n + 1 ) ! m ( m − 1 ) ⋯ ( m − n + 1 ) x n + 1 ( 1 + ξ ) m − n − 1
或
(
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
( 1 + x ) m = 1 + m x + 2 ! m ( m − 1 ) x 2 + ⋯ ,
+
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}})
+ n ! m ( m − 1 ) ⋯ ( m − n + 1 ) x n + o ( x n )
12.函数单调性的判断 Th1: 设函数
f
(
x
)
f(x)
f ( x ) 在
(
a
,
b
)
(a,b)
( a , b ) 区间内可导,如果对
∀
x
∈
(
a
,
b
)
\forall x\in (a,b)
∀ x ∈ ( a , b ) ,都有
f
′
(
x
)
>
0
f\,'(x)>0
f ′ ( x ) > 0 (或
f
′
(
x
)
<
0
f\,'(x)<0
f ′ ( x ) < 0 ),则函数
f
(
x
)
f(x)
f ( x ) 在
(
a
,
b
)
(a,b)
( a , b ) 内是单调增加的(或单调减少)
Th2: (取极值的必要条件)设函数
f
(
x
)
f(x)
f ( x ) 在
x
0
{{x}_{0}}
x 0 处可导,且在
x
0
{{x}_{0}}
x 0 处取极值,则
f
′
(
x
0
)
=
0
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)
f ( x ) 在
x
0
{{x}_{0}}
x 0 的某一邻域内可微,且
f
′
(
x
0
)
=
0
f\,'({{x}_{0}})=0
f ′ ( x 0 ) = 0 (或
f
(
x
)
f(x)
f ( x ) 在
x
0
{{x}_{0}}
x 0 处连续,但
f
′
(
x
0
)
f\,'({{x}_{0}})
f ′ ( x 0 ) 不存在。) (1)若当
x
x
x 经过
x
0
{{x}_{0}}
x 0 时,
f
′
(
x
)
f\,'(x)
f ′ ( x ) 由“+”变“-”,则
f
(
x
0
)
f({{x}_{0}})
f ( x 0 ) 为极大值; (2)若当
x
x
x 经过
x
0
{{x}_{0}}
x 0 时,
f
′
(
x
)
f\,'(x)
f ′ ( x ) 由“-”变“+”,则
f
(
x
0
)
f({{x}_{0}})
f ( x 0 ) 为极小值; (3)若
f
′
(
x
)
f\,'(x)
f ′ ( x ) 经过
x
=
x
0
x={{x}_{0}}
x = x 0 的两侧不变号,则
f
(
x
0
)
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)
f ( x ) 在点
x
0
{{x}_{0}}
x 0 处有
f
′
′
(
x
)
≠
0
f''(x)\ne 0
f ′ ′ ( x ) = 0 ,且
f
′
(
x
0
)
=
0
f\,'({{x}_{0}})=0
f ′ ( x 0 ) = 0 ,则 当
f
′
′
(
x
0
)
<
0
f'\,'({{x}_{0}})<0
f ′ ′ ( x 0 ) < 0 时,
f
(
x
0
)
f({{x}_{0}})
f ( x 0 ) 为极大值; 当
f
′
′
(
x
0
)
>
0
f'\,'({{x}_{0}})>0
f ′ ′ ( x 0 ) > 0 时,
f
(
x
0
)
f({{x}_{0}})
f ( x 0 ) 为极小值。 注:如果
f
′
′
(
x
0
)
<
0
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
x → + ∞ lim f ( x ) = b ,或
lim
x
→
−
∞
f
(
x
)
=
b
\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=b
x → − ∞ lim f ( x ) = b ,则
y
=
b
y=b
y = b 称为函数
y
=
f
(
x
)
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
x → x 0 − lim f ( x ) = ∞ ,或
lim
x
→
x
0
+
f
(
x
)
=
∞
\underset{x\to x_{0}^{+}}{\mathop{\lim }}\,f(x)=\infty
x → x 0 + lim f ( x ) = ∞ ,则
x
=
x
0
x={{x}_{0}}
x = x 0 称为
y
=
f
(
x
)
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]
a = x → ∞ lim x f ( x ) , b = x → ∞ lim [ f ( x ) − a x ] ,则
y
=
a
x
+
b
y=ax+b
y = a x + b 称为
y
=
f
(
x
)
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
f ′ ′ ( x ) < 0 (或
f
′
′
(
x
)
>
0
f''(x)>0
f ′ ′ ( x ) > 0 ),则
f
(
x
)
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}}
x 0 处
f
′
′
(
x
)
=
0
f''(x)=0
f ′ ′ ( x ) = 0 ,(或
f
′
′
(
x
)
f''(x)
f ′ ′ ( x ) 不存在),当
x
x
x 变动经过
x
0
{{x}_{0}}
x 0 时,
f
′
′
(
x
)
f''(x)
f ′ ′ ( x ) 变号,则
(
x
0
,
f
(
x
0
)
)
({{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)
f ( x ) 在
x
0
{{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
)
≠
0
f'''(x)\ne 0
f ′ ′ ′ ( x ) = 0 ,则
(
x
0
,
f
(
x
0
)
)
({{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
d S = 1 + y ′ 2
d x
16.曲率
曲线
y
=
f
(
x
)
y=f(x)
y = f ( x ) 在点
(
x
,
y
)
(x,y)
( x , y ) 处的曲率
k
=
∣
y
′
′
∣
(
1
+
y
′
2
)
3
2
k=\frac{\left| y'' \right|}{{{(1+y{{'}^{2}})}^{\tfrac{3}{2}}}}
k = ( 1 + y ′ 2 ) 2 3 ∣ y ′ ′ ∣ 。 对于参数方程KaTeX parse error: No such environment: align at position 15: \left\{ \begin{̲a̲l̲i̲g̲n̲}̲ & x=\varphi (…
k
=
∣
φ
′
(
t
)
ψ
′
′
(
t
)
−
φ
′
′
(
t
)
ψ
′
(
t
)
∣
[
φ
′
2
(
t
)
+
ψ
′
2
(
t
)
]
3
2
k=\frac{\left| \varphi '(t)\psi ''(t)-\varphi ''(t)\psi '(t) \right|}{{{[\varphi {{'}^{2}}(t)+\psi {{'}^{2}}(t)]}^{\tfrac{3}{2}}}}
k = [ φ ′ 2 ( t ) + ψ ′ 2 ( t ) ] 2 3 ∣ φ ′ ( t ) ψ ′ ′ ( t ) − φ ′ ′ ( t ) ψ ′ ( t ) ∣ 。
17.曲率半径
曲线在点
M
M
M 处的曲率
k
(
k
≠
0
)
k(k\ne 0)
k ( k = 0 ) 与曲线在点
M
M
M 处的曲率半径
ρ
\rho
ρ 有如下关系:
ρ
=
1
k
\rho =\frac{1}{k}
ρ = k 1 。
参考:https://aistudio.baidu.com/aistudio/projectdetail/549642