有界与最值定理:
m
≤
f
(
x
)
≤
M
m \leq f(x)\leq M
m≤f(x)≤M
介值定理:
m
≤
μ
≤
M
,
∃
ϵ
∈
[
a
,
b
]
,
f
(
ϵ
)
=
μ
m\leq \mu \leq M, \exists \epsilon \in[a,b], f(\epsilon) = \mu
m≤μ≤M,∃ϵ∈[a,b],f(ϵ)=μ
平均值定理:
a
<
x
1
<
x
2
<
⋯
<
x
n
<
b
,
∃
ϵ
∈
[
x
1
,
x
n
]
,
f
(
ϵ
)
=
f
(
x
1
)
+
f
(
x
2
)
+
⋯
+
f
(
x
n
)
n
a<x_1<x_2<\cdots<x_n<b, \exists \epsilon \in[x_1,x_n], f(\epsilon) = \frac{f(x_1)+f(x_2)+\cdots+f(x_n)}{n}
a<x1<x2<⋯<xn<b,∃ϵ∈[x1,xn],f(ϵ)=nf(x1)+f(x2)+⋯+f(xn)
零点定理:
f
(
a
)
⋅
f
(
b
)
<
0
,
∃
ϵ
∈
[
a
,
b
]
,
f
(
ϵ
)
=
0
f(a)\cdot f(b)<0, \exists \epsilon \in[a,b], f(\epsilon) = 0
f(a)⋅f(b)<0,∃ϵ∈[a,b],f(ϵ)=0
费马定理:
x
0
处
可
导
且
为
极
值
,
f
′
(
x
0
)
=
0
x_0处可导且为极值, f'(x_0) = 0
x0处可导且为极值,f′(x0)=0
罗尔定理:
[
a
,
b
)
可
导
,
f
(
a
)
=
f
(
b
)
,
∃
ϵ
∈
[
a
,
b
]
,
f
′
(
ϵ
)
=
0
[a,b)可导, f(a) = f(b), \exists \epsilon \in[a,b], f'(\epsilon) = 0
[a,b)可导,f(a)=f(b),∃ϵ∈[a,b],f′(ϵ)=0
拉格朗日中值定理:
(
a
,
b
)
可
导
,
∃
ϵ
∈
[
a
,
b
]
,
f
(
b
)
−
f
(
a
)
=
f
′
(
ϵ
)
(
b
−
a
)
(a,b)可导, \exists \epsilon \in[a,b], f(b) - f(a) = f'(\epsilon)(b-a)
(a,b)可导,∃ϵ∈[a,b],f(b)−f(a)=f′(ϵ)(b−a)
柯西中值定理:
(
a
,
b
)
可
导
,
g
′
(
x
)
≠
0
,
∃
ϵ
∈
[
a
,
b
]
,
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
=
f
′
(
ϵ
)
g
′
(
ϵ
)
(a,b)可导, g'(x)\neq 0, \exists \epsilon \in[a,b], \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(\epsilon)}{g'(\epsilon)}
(a,b)可导,g′(x)=0,∃ϵ∈[a,b],g(b)−g(a)f(b)−f(a)=g′(ϵ)f′(ϵ)
泰勒公式(拉格朗日余项):
f
(
x
)
=
f
(
x
0
)
+
f
′
(
x
0
)
(
x
−
x
0
)
+
⋯
+
f
(
n
)
(
x
0
)
n
!
(
x
−
x
0
)
n
+
f
(
n
+
1
)
(
ϵ
)
(
n
+
1
)
!
(
x
−
x
0
)
n
+
1
f(x) = f(x_0) + f'(x_0)(x-x_0)+\cdots+ \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n +\frac{f^{(n+1)}(\epsilon)}{(n+1)!}(x-x_0)^{n+1}
f(x)=f(x0)+f′(x0)(x−x0)+⋯+n!f(n)(x0)(x−x0)n+(n+1)!f(n+1)(ϵ)(x−x0)n+1
泰勒公式(佩亚诺余项):
f
(
x
)
=
f
(
x
0
)
+
f
′
(
x
0
)
(
x
−
x
0
)
+
⋯
+
f
(
n
)
(
x
0
)
n
!
(
x
−
x
0
)
n
+
O
(
(
x
−
x
0
)
n
)
f(x) = f(x_0) + f'(x_0)(x-x_0)+\cdots+ \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n +O((x-x_0)^n)
f(x)=f(x0)+f′(x0)(x−x0)+⋯+n!f(n)(x0)(x−x0)n+O((x−x0)n)
积分中值定理:
∃
ϵ
∈
[
a
,
b
]
,
∫
a
b
f
(
x
)
d
x
b
−
a
=
f
(
ϵ
)
\exists \epsilon \in[a,b], \frac{\int_a^bf(x)dx}{b-a} = f(\epsilon)
∃ϵ∈[a,b],b−a∫abf(x)dx=f(ϵ)