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高等数学
伯努利不等式
(
1
+
x
)
n
≥
1
+
n
x
(
x
>
−
1
,
n
∈
N
+
)
(1 + x)^n \geq 1 + nx \quad (x > -1, \ n \in \mathbb{N}^+ )
(
1
+
x
)
n
≥
1
+
n
x
(
x
>
−
1
,
n
∈
N
+
)
二次不等式（判别式）
a
x
2
+
b
x
+
c
≥
0
或
a
x
2
+
b
x
+
c
≤
0
ax^2 + bx + c \geq 0 \text{ 或 } ax^2 + bx + c \leq 0
a
x
2
+
b
x
+
c
≥
0
或
a
x
2
+
b
x
+
c
≤
0
柯西不等式（均方根不等式）
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
(
b
1
2
+
b
2
2
+
⋯
+
b
n
2
)
≥
(
a
1
b
1
+
a
2
b
2
+
⋯
+
a
n
b
n
)
2
(a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2) \geq (a_1b_1 + a_2b_2 + \cdots + a_nb_n)^2
(
a
1
2
+
a
2
2
+
⋯
+
a
n
2
)
(
b
1
2
+
b
2
2
+
⋯
+
b
n
2
)
≥
(
a
1
b
1
+
a
2
b
2
+
⋯
+
a
n
b
n
)
2
洛必达法则（极限不等式）
lim
x
→
a
f
(
x
)
g
(
x
)
=
lim
x
→
a
f
′
(
x
)
g
′
(
x
)
当
g
′
(
x
)
≠
0
\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{当 } g'(x) \neq 0
lim
x
→
a
g
(
x
)
f
(
x
)
=
lim
x
→
a
g
′
(
x
)
f
′
(
x
)
当
g
′
(
x
)
=
0
闵科夫斯基不等式
(
∑
i
=
1
n
(
a
i
+
b
i
)
p
)
1
/
p
≤
(
∑
i
=
1
n
a
i
p
)
1
/
p
+
(
∑
i
=
1
n
b
i
p
)
1
/
p
( \sum_{i=1}^n (a_i + b_i)^p )^{1/p} \leq ( \sum_{i=1}^n a_i^p )^{1/p} + ( \sum_{i=1}^n b_i^p )^{1/p}
(
∑
i
=
1
n
(
a
i
+
b
i
)
p
)
1
/
p
≤
(
∑
i
=
1
n
a
i
p
)
1
/
p
+
(
∑
i
=
1
n
b
i
p
)
1
/
p
切比雪夫不等式
a
1
b
1
+
a
2
b
2
+
⋯
+
a
n
b
n
≥
a
1
+
a
2
+
⋯
+
a
n
b
1
+
b
2
+
⋯
+
b
n
\frac{a_1}{b_1} + \frac{a_2}{b_2} + \cdots + \frac{a_n}{b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{b_1 + b_2 + \cdots + b_n}
b
1
a
1
+
b
2
a
2
+
⋯
+
b
n
a
n
≥
b
1
+
b
2
+
⋯
+
b
n
a
1
+
a
2
+
⋯
+
a
n
三角形不等式（向量形式）
∣
a
+
b
∣
≤
∣
a
∣
+
∣
b
∣
| \mathbf{a} + \mathbf{b} | \leq | \mathbf{a} | + | \mathbf{b} |
∣
a
+
b
∣
≤
∣
a
∣
+
∣
b
∣
斯特林公式
n
!
≈
2
π
n
(
n
e
)
n
n! \approx \sqrt{2\pi n} \left( \frac{n}{e} \right)^n
n
!
≈
2
π
n
(
e
n
)
n
算术-几何-调和平均值不等式
a
+
b
2
≥
a
b
≥
2
a
b
a
+
b
\frac{a + b}{2} \geq \sqrt{ab} \geq \frac{2ab}{a+b}
2
a
+
b
≥
a
b
≥
a
+
b
2
a
b
算术平均值不小于几何平均值
a
1
+
a
2
+
⋯
+
a
n
n
≥
a
1
a
2
⋯
a
n
n
\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}
n
a
1
+
a
2
+
⋯
+
a
n
≥
n
a
1
a
2
⋯
a
n
詹森不等式
f
(
a
1
+
a
2
+
⋯
+
a
n
n
)
≤
f
(
a
1
)
+
f
(
a
2
)
+
⋯
+
f
(
a
n
)
n
f\left( \frac{a_1 + a_2 + \cdots + a_n}{n} \right) \leq \frac{f(a_1) + f(a_2) + \cdots + f(a_n)}{n}
f
(
n
a
1
+
a
2
+
⋯
+
a
n
)
≤
n
f
(
a
1
)
+
f
(
a
2
)
+
⋯
+
f
(
a
n
)
正数的加权平均值不等关系
a
1
w
1
+
a
2
w
2
+
⋯
+
a
n
w
n
w
1
+
w
2
+
⋯
+
w
n
≥
a
1
w
1
a
2
w
2
⋯
a
n
w
n
n
\frac{a_1 w_1 + a_2 w_2 + \cdots + a_n w_n}{w_1 + w_2 + \cdots + w_n} \geq \sqrt[n]{a_1^{w_1} a_2^{w_2} \cdots a_n^{w_n}}
w
1
+
w
2
+
⋯
+
w
n
a
1
w
1
+
a
2
w
2
+
⋯
+
a
n
w
n
≥
n
a
1
w
1
a
2
w
2
⋯
a
n
w
n
对数函数
y
=
log
a
x
y = \log_a{x}
y
=
lo
g
a
x
二次函数
y
=
a
x
2
+
b
x
+
c
y = ax^2 + bx + c
y
=
a
x
2
+
b
x
+
c
反比例函数
y
=
k
x
y = \frac{k}{x}
y
=
x
k
一次函数
y
=
m
x
+
b
y = mx + b
y
=
m
x
+
b
指数函数
y
=
a
x
y = a^x
y
=
a
x
导数定义
f
′
(
x
)
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
f
′
(
x
)
=
lim
h
→
0
h
f
(
x
+
h
)
−
f
(
x
)
对数函数导数
d
d
x
(
log
a
x
)
=
1
x
ln
a
\frac{d}{dx}(\log_a{x}) = \frac{1}{x \ln a}
d
x
d
(
lo
g
a
x
)
=
x
ln
a
1
复合函数导数
d
d
x
(
f
(
g
(
x
)
)
)
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)
d
x
d
(
f
(
g
(
x
)
)
)
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
幂函数导数
d
d
x
(
x
n
)
=
n
x
n
−
1
\frac{d}{dx}(x^n) = nx^{n-1}
d
x
d
(
x
n
)
=
n
x
n
−
1
指数函数导数
d
d
x
(
e
x
)
=
e
x
\frac{d}{dx}(e^x) = e^x
d
x
d
(
e
x
)
=
e
x
复数方程
z
=
a
+
b
i
z = a + bi
z
=
a
+
b
i
高次方程
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
0
=
0
x^n + a_{n-1}x^{n-1} + ... + a_0 = 0
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
0
=
0
线性方程组
{
a
x
+
b
y
=
c
d
x
+
e
y
=
f
\begin{cases} ax + by = c \\ dx + ey = f \end{cases}
{
a
x
+
b
y
=
c
d
x
+
e
y
=
f
一元二次方程
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
对称轴
x
=
−
b
2
a
x = -\frac{b}{2a}
x
=
−
2
a
b
方程的解
x
=
−
b
±
b
2
−
4
a
c
2
a
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
x
=
2
a
−
b
±
b
2
−
4
a
c
根的和
x
1
+
x
2
=
−
b
a
x_1 + x_2 = -\frac{b}{a}
x
1
+
x
2
=
−
a
b
根的积
x
1
⋅
x
2
=
c
a
x_1 \cdot x_2 = \frac{c}{a}
x
1
⋅
x
2
=
a
c
线性方程
a
x
+
b
y
=
c
ax + by = c
a
x
+
b
y
=
c
等比数列
a
n
=
a
1
⋅
r
n
−
1
a_n = a_1 \cdot r^{n-1}
a
n
=
a
1
⋅
r
n
−
1
等差数列
a
n
=
a
1
+
(
n
−
1
)
d
a_n = a_1 + (n-1)d
a
n
=
a
1
+
(
n
−
1
)
d
等差数列求和
S
n
=
n
2
(
a
1
+
a
n
)
S_n = \frac{n}{2} (a_1 + a_n)
S
n
=
2
n
(
a
1
+
a
n
)
矩形的面积
S
=
a
×
b
S = a \times b
S
=
a
×
b
平行四边形的面积
S
=
b
×
h
S = b \times h
S
=
b
×
h
三角形的面积
S
=
1
2
×
b
×
h
S = \frac{1}{2} \times b \times h
S
=
2
1
×
b
×
h
圆的面积
S
=
π
r
2
S = \pi r^2
S
=
π
r
2
圆的周长
C
=
2
π
r
C = 2\pi r
C
=
2
π
r
直角三角形的勾股定理
a
2
+
b
2
=
c
2
a^2 + b^2 = c^2
a
2
+
b
2
=
c
2
向量的夹角
θ
=
cos
−
1
(
u
⃗
⋅
v
⃗
∣
u
⃗
∣
∣
v
⃗
∣
)
\theta = \cos^{-1} \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \right)
θ
=
cos
−
1
(
∣
u
∣
∣
v
∣
u
⋅
v
)
向量的模
∣
v
⃗
∣
=
v
1
2
+
v
2
2
+
v
3
2
+
⋯
+
v
n
2
|\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2 + \cdots + v_n^2}
∣
v
∣
=
v
1
2
+
v
2
2
+
v
3
2
+
⋯
+
v
n
2
向量的数量积
u
⃗
⋅
v
⃗
=
∣
u
⃗
∣
∣
v
⃗
∣
cos
(
θ
)
\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos(\theta)
u
⋅
v
=
∣
u
∣
∣
v
∣
cos
(
θ
)
向量定义
a
⃗
=
(
x
,
y
)
\vec{a} = (x, y)
a
=
(
x
,
y
)
数量积的计算
u
⃗
⋅
v
⃗
=
u
1
v
1
+
u
2
v
2
+
u
3
v
3
+
⋯
+
u
n
v
n
\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 + \cdots + u_n v_n
u
⋅
v
=
u
1
v
1
+
u
2
v
2
+
u
3
v
3
+
⋯
+
u
n
v
n
向量加法
a
⃗
+
b
⃗
=
(
x
1
+
x
2
,
y
1
+
y
2
)
\vec{a} + \vec{b} = (x_1 + x_2, y_1 + y_2)
a
+
b
=
(
x
1
+
x
2
,
y
1
+
y
2
)
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