常用数学公式和定理 Last Updated: 2024-03-23 14:49:37 Saturday
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本文汇总常用的数学公式和定理,这些基本都是中学数学的内容,是进行有效的高等数学学习的重要基础。欢迎程序员同学收藏!
A function f ( n ) f(n) f ( n ) is monotonically increasing if m ≤ n m\le n m ≤ n implies f ( m ) ≤ f ( n ) f(m)\le f(n) f ( m ) ≤ f ( n ) . Similarly, it is monotonically decreasing if m ≤ n m\le n m ≤ n implies f ( m ) ≥ f ( n ) f(m)\ge f(n) f ( m ) ≥ f ( n ) . A function f ( n ) f(n) f ( n ) is strictly increasing if m < n m\lt n m < n implies f ( m ) < f ( n ) f(m)\lt f(n) f ( m ) < f ( n ) and strictly decreasing if m < n m\lt n m < n implies f ( m ) > f ( n ) f(m)\gt f(n) f ( m ) > f ( n ) .
a m a n = a m + n ( a m ) n = a m n ( a b ) m = a m ⋅ b m a m a n = a m − n , ( if n ≠ 0 , a ≠ 0 ) x − a = 1 x a , ( if a ≠ 0 , x ≠ 0 ) x 0 = 1 a m n = a m n , ( n ≠ 0 ) x n + x n = 2 ⋅ x n ≠ x 2 n 2 n + 2 n = 2 n + 1 \begin{align}
a^ma^n &= a^{m+n}\notag \\
(a^m)^n &= a^{mn}\notag \\
(ab)^m &= a^m{\cdot}b^m\notag \\
\frac{a^m}{a^n} &= a^{m-n},(\text{if }n\neq0, a\neq0)\notag \\
x^{-a} &= \cfrac{1}{x^a},(\text{if }a\neq0, x\neq0)\notag \\
x^0 &= 1\notag \\
a^{\frac{m}{n}} &= \sqrt[n]{a^m},(n\neq0)\notag \\
x^n + x^n &= 2\cdot x^n \neq x^{2n}\notag \\
2^n + 2^n &= 2^{n+1}\notag
\end{align} a m a n ( a m ) n ( ab ) m a n a m x − a x 0 a n m x n + x n 2 n + 2 n = a m + n = a mn = a m ⋅ b m = a m − n , ( if n = 0 , a = 0 ) = x a 1 , ( if a = 0 , x = 0 ) = 1 = n a m , ( n = 0 ) = 2 ⋅ x n = x 2 n = 2 n + 1
对零次方的推导:
x a ⋅ x − a = x a − a = x a x a = 1 = x 0 x^a\cdot x^{-a}=x^{a-a}=\cfrac{x^a}{x^a}=1=x^0 x a ⋅ x − a = x a − a = x a x a = 1 = x 0
任何数的0次方都得1,包括0
1的任何次方都得1,包括0
分数次幂就是开根号:
( x 1 n ) n = x ⟹ x 1 n = x n (x^{\frac{1}{n}})^n=x \Longrightarrow x^{\frac{1}{n}}=\sqrt[n]{x} ( x n 1 ) n = x ⟹ x n 1 = n x
幂函数y = a x y=a^x y = a x 会要求a > 0 , a ≠ 1 a\gt0,a\neq1 a > 0 , a = 1 ,有y > 0 y\gt0 y > 0
对数计算的底数a > 0 , a ≠ 1 a\gt0,a\neq1 a > 0 , a = 1 ,与幂函数的规定对应。
log a x − log a y = log a x y log a x + log a y = log a x y log a b n = n ⋅ log a b log a m b = 1 m ⋅ log a b log a b = log a b log a a = log c b log c a a log b c = c log b a log 2 x < x , ∀ x > 0 \begin{align}
\log_ax - \log_ay &= \log_a\frac{x}{y}\notag \\
\log_ax + \log_ay &= \log_a{xy}\notag \\
\log_a{b^n} &= n\cdot\log_ab\notag \\
\log_{a^m}{b} &= \frac{1}{m}\cdot\log_ab\notag \\
\log_ab =\frac{\log_ab}{\log_aa}&=\frac{\log_cb}{\log_ca}\notag \\
a^{\log_bc} &= c^{\log_ba}\notag \\
\log_2{x} &< x,\ \forall x>0 \notag \\
\end{align} log a x − log a y log a x + log a y log a b n log a m b log a b = log a a log a b a l o g b c log 2 x = log a y x = log a x y = n ⋅ log a b = m 1 ⋅ log a b = log c a log c b = c l o g b a < x , ∀ x > 0
对数函数y = log a x y=\log_ax y = log a x 规定a > 0 , a ≠ 1 a\gt0,a\neq1 a > 0 , a = 1 ,有x > 0 x\gt0 x > 0
a 2 − b 2 = ( a + b ) ⋅ ( a − b ) ( a ± b ) 2 = a 2 ± 2 a b + b 2 ( a ± b ) 3 = a 3 ± 3 a 2 b + 3 a b 2 ± b 3 1 ± x 3 = ( 1 ± x ) ( 1 ∓ x + x 2 ) a 3 ± b 3 = ( a ± b ) ( a ∓ a b + b 2 ) ( a + b ) u = a u ( 1 + b a ) u = b u ( a b + 1 ) u \begin{align}
a^2 - b^2 &= (a+b)\cdot(a-b)\notag \\
(a \pm b)^2 &= a^2 \pm 2ab + b^2\notag \\
(a \pm b)^3 &= a^3 \pm 3a^2b + 3ab^2 \pm b^3\notag \\
1 \pm x^3 &= (1 \pm x)(1 \mp x+x^2)\notag \\
a^3 \pm b^3 &= (a \pm b)(a \mp ab+b^2)\notag \\
(a + b)^u &= a^u(1+\frac{b}{a})^u\notag \\
&= b^u(\frac{a}{b} +1)^u\notag
\end{align} a 2 − b 2 ( a ± b ) 2 ( a ± b ) 3 1 ± x 3 a 3 ± b 3 ( a + b ) u = ( a + b ) ⋅ ( a − b ) = a 2 ± 2 ab + b 2 = a 3 ± 3 a 2 b + 3 a b 2 ± b 3 = ( 1 ± x ) ( 1 ∓ x + x 2 ) = ( a ± b ) ( a ∓ ab + b 2 ) = a u ( 1 + a b ) u = b u ( b a + 1 ) u
( a ± b ) n = ∑ r = 0 n C n r ⋅ ( − 1 ) r ⋅ a n − r ⋅ b r (a \pm b)^n = \sum_{r=0}^{n}C_n^r \cdot (-1)^r \cdot a^{n-r} \cdot b^r ( a ± b ) n = r = 0 ∑ n C n r ⋅ ( − 1 ) r ⋅ a n − r ⋅ b r
C n r C_n^r C n r 是一个组合 数。r r r 可以理解为b的幂次。
把二项式的系数按n从小到大取值,每个n一行,堆在一起,就是杨辉三角
。
[ f ( x ) ] g ( x ) = e g ( x ) ⋅ ln f ( x ) , ( f ( x ) > 0 ) [f(x)]^{g(x)} = e^{g(x)\cdot\ln{f(x)}},\
(f(x)>0) [ f ( x ) ] g ( x ) = e g ( x ) ⋅ l n f ( x ) , ( f ( x ) > 0 )
sin 2 α + cos 2 α = 1 1 + tan 2 α = sec 2 α 1 + cot 2 α = csc 2 α sin ( a + b ) = sin a cos b + cos a sin b sin ( a − b ) = sin a cos b − cos a sin b cos ( a + b ) = cos a cos b − sin a sin b cos ( a − b ) = cos a cos b + sin a sin b sin a + sin b = 2 ⋅ sin a + b 2 ⋅ cos a − b 2 sin a − sin b = 2 ⋅ cos a + b 2 ⋅ sin a − b 2 cos a + cos b = 2 ⋅ cos a + b 2 ⋅ cos a − b 2 sin a − sin b = − 2 ⋅ sin a + b 2 ⋅ sin a − b 2 tan a + tan b = sin ( a + b ) cos a ⋅ cos b 1 − cos 2 x = 2 ⋅ sin 2 x 1 + cos 2 x = 2 ⋅ cos 2 x 1 ± sin 2 x = ( sin x ± cos x ) 2 \begin{aligned}
\sin^2\alpha + \cos^2\alpha &= 1 \\
1 + \tan^2\alpha &= \sec^2\alpha \\
1 + \cot^2\alpha &= \csc^2\alpha \\
\sin(a+b) &= \sin{a}\cos{b}+\cos{a}\sin{b}\\
\sin(a-b) &= \sin{a}\cos{b}-\cos{a}\sin{b} \\
\cos(a+b) &= \cos{a}\cos{b}-\sin{a}\sin{b} \\
\cos(a-b) &= \cos{a}\cos{b}+\sin{a}\sin{b} \\
\sin{a}+\sin{b} &= 2{\cdot}\sin{\frac{a+b}{2}}{\cdot}\cos{\frac{a-b}{2}} \\
\sin{a}-\sin{b} &= 2{\cdot}\cos{\frac{a+b}{2}}{\cdot}\sin{\frac{a-b}{2}} \\
\cos{a}+\cos{b} &= 2{\cdot}\cos{\frac{a+b}{2}}{\cdot}\cos{\frac{a-b}{2}} \\
\sin{a}-\sin{b} &= -2{\cdot}\sin{\frac{a+b}{2}}{\cdot}\sin{\frac{a-b}{2}} \\
\tan{a}+\tan{b} &= \frac{\sin{(a+b)}}{\cos{a}{\cdot}\cos{b}} \\
1-\cos{2x} &= 2{\cdot}\sin^2{x} \\
1+\cos{2x} &= 2{\cdot}\cos^2{x} \\
1 \pm \sin{2x} &= (\sin{x} \pm \cos{x})^2
\end{aligned} sin 2 α + cos 2 α 1 + tan 2 α 1 + cot 2 α sin ( a + b ) sin ( a − b ) cos ( a + b ) cos ( a − b ) sin a + sin b sin a − sin b cos a + cos b sin a − sin b tan a + tan b 1 − cos 2 x 1 + cos 2 x 1 ± sin 2 x = 1 = sec 2 α = csc 2 α = sin a cos b + cos a sin b = sin a cos b − cos a sin b = cos a cos b − sin a sin b = cos a cos b + sin a sin b = 2 ⋅ sin 2 a + b ⋅ cos 2 a − b = 2 ⋅ cos 2 a + b ⋅ sin 2 a − b = 2 ⋅ cos 2 a + b ⋅ cos 2 a − b = − 2 ⋅ sin 2 a + b ⋅ sin 2 a − b = cos a ⋅ cos b sin ( a + b ) = 2 ⋅ sin 2 x = 2 ⋅ cos 2 x = ( sin x ± cos x ) 2
sin ( a + b ) \sin(a+b) sin ( a + b ) 这个公式可以说最重要,但是它很好记忆。它下面的哪些公式,都可以用这个(*)来推导,具体可参考:和差化积公式的推导,二倍角公式的推导。
a x 2 + b x + c = 0 , Δ = b 2 − 4 a c ax^2+bx+c=0,\Delta=b^2-4ac a x 2 + b x + c = 0 , Δ = b 2 − 4 a c ,当 Δ < 0 \Delta\lt0 Δ < 0 时,无实数根,
求根公式:x = − b ± Δ 2 a x=\cfrac{-b\pm\sqrt{\Delta}}{2a} x = 2 a − b ± Δ
a > 0 a>0 a > 0 时,函数图像开口向上;
a < 0 a<0 a < 0 时,函数图像开口向下;
Δ < 0 \Delta < 0 Δ < 0 时,函数图像与x轴没有交点。
推导:
a x 2 + b x + c = a ( x 2 + b a x + c a ) = a ( x 2 + b a x + b 2 4 a 2 + c a − b 2 4 a 2 ) = a ( ( x + b 2 a ) 2 + 4 a c − b 2 4 a 2 ) = 0 ax^2+bx+c \\
= a(x^2+\frac{b}{a}x+\frac{c}{a}) \\
= a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2}+\frac{c}{a}-\frac{b^2}{4a^2}) \\
= a((x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a^2})\\
= 0 a x 2 + b x + c = a ( x 2 + a b x + a c ) = a ( x 2 + a b x + 4 a 2 b 2 + a c − 4 a 2 b 2 ) = a (( x + 2 a b ) 2 + 4 a 2 4 a c − b 2 ) = 0
然后就可以解出x了!
韦达定理应用于一元二次方程
韦达定理说明了一元方程中根和系数之间的关系。法国数学家弗朗索瓦·韦达(F. Vieta,1540—1603)于1615年在著作《论方程的识别与订正》中建立了方程根与系数的关系,提出了这条定理。由于韦达最早发现代数方程的根与系数之间有这种关系,人们把这个关系称为韦达定理。
当一元二次方程存在两个根时:
{ x 1 + x 2 = − b a x 1 x 2 = c a \begin{cases}
x_1+x_2=-\cfrac{b}{a} \\
x_1x_2=\cfrac{c}{a} \\
\end{cases} ⎩ ⎨ ⎧ x 1 + x 2 = − a b x 1 x 2 = a c
总结4种平均数
数列和
在一个不等式中,如果不论用任何实数代入该不等式,它都是成立的,那么这样的不等式叫作绝对不等式。比如:r 2 ≥ 0 r^2 \ge 0 r 2 ≥ 0 。
(1) ∣ a ∣ − ∣ b ∣ ≤ ∣ a − b ∣ ≤ ∣ a ∣ + ∣ b ∣ |a|-|b| \le |a-b| \le |a|+|b| ∣ a ∣ − ∣ b ∣ ≤ ∣ a − b ∣ ≤ ∣ a ∣ + ∣ b ∣
∣ a − b ∣ |a-b| ∣ a − b ∣ 表示两点间的距离,与 ∣ b − a ∣ |b-a| ∣ b − a ∣ 相等。
(2) ∣ x + y ∣ ≤ ∣ x ∣ + ∣ y ∣ |x+y| \le |x|+|y| ∣ x + y ∣ ≤ ∣ x ∣ + ∣ y ∣
仔细想想,以上字母代表的数字都在R内,加和减还有什么区别呢,因此:
(3) ∣ a ∣ − ∣ b ∣ ≤ ∣ a ± b ∣ ≤ ∣ a ∣ + ∣ b ∣ |a|-|b| \le |a \pm b| \le |a|+|b| ∣ a ∣ − ∣ b ∣ ≤ ∣ a ± b ∣ ≤ ∣ a ∣ + ∣ b ∣
其实:∣ ∣ a ∣ − ∣ b ∣ ∣ ≤ ∣ a ± b ∣ ≤ ∣ a ∣ + ∣ b ∣ ||a|-|b|| \le |a \pm b| \le |a|+|b| ∣∣ a ∣ − ∣ b ∣∣ ≤ ∣ a ± b ∣ ≤ ∣ a ∣ + ∣ b ∣
(4) 三角不等式: ∣ a − c ∣ ≤ ∣ a − b ∣ + ∣ b − c ∣ |a-c| \le |a-b| + |b-c| ∣ a − c ∣ ≤ ∣ a − b ∣ + ∣ b − c ∣
(5) a为圆心,r为半径:∣ x − a ∣ ≤ r ⟺ ( a − r ) ≤ x ≤ ( a + r ) |x-a| \le r \iff (a-r) \le x \le (a+r) ∣ x − a ∣ ≤ r ⟺ ( a − r ) ≤ x ≤ ( a + r )
推导:
∣ x ∣ = ∣ x − y + y ∣ ≤ ∣ x − y ∣ + ∣ y ∣ |x| = |x-y+y| \le |x-y|+|y| ∣ x ∣ = ∣ x − y + y ∣ ≤ ∣ x − y ∣ + ∣ y ∣ 得 ∣ x ∣ − ∣ y ∣ ≤ ∣ x − y ∣ |x|-|y| \le |x-y| ∣ x ∣ − ∣ y ∣ ≤ ∣ x − y ∣
∣ x ∣ = ∣ x + y − y ∣ ≤ ∣ x + y ∣ + ∣ y ∣ |x| = |x+y-y| \le |x+y|+|y| ∣ x ∣ = ∣ x + y − y ∣ ≤ ∣ x + y ∣ + ∣ y ∣ 得 ∣ x ∣ − ∣ y ∣ ≤ ∣ x + y ∣ |x|-|y| \le |x+y| ∣ x ∣ − ∣ y ∣ ≤ ∣ x + y ∣
当x ≥ 0 , y ≥ 0 x\ge0,y\ge0 x ≥ 0 , y ≥ 0 时,算术平均数大于等于几何平均数,当x = y x=y x = y 时取等号:
x + y 2 ≥ x y \cfrac{x+y}{2}\ge\sqrt{xy} 2 x + y ≥ x y
基本不等式的一般形式,x i ≥ 0 x_i\ge0 x i ≥ 0 :
( ∑ i = 1 n x i ) ⋅ 1 n ≥ ( ∏ i = 1 n x i ) 1 n \left(\sum_{i=1}^n x_i\right)\cdot\cfrac{1}{n}\ge\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}} ( i = 1 ∑ n x i ) ⋅ n 1 ≥ ( i = 1 ∏ n x i ) n 1
设扇形对应的源半径为r r r ,扇形弧长为l l l ,其面积公式为:
π r 2 ⋅ l 2 π r = 1 2 ⋅ r ⋅ l \pi r^2 \cdot \frac{l}{2\pi r} = \frac{1}{2} \cdot r\cdot l π r 2 ⋅ 2 π r l = 2 1 ⋅ r ⋅ l
弧度与弧长的关系:
2 π r 2\pi r 2 π r 的弧长对应 2 π 2\pi 2 π 弧度,因此任何弧度θ \theta θ 与对应弧长l l l 的关系是:
l 2 π r = θ 2 π \frac{l}{2\pi r} = \frac{\theta}{2\pi} 2 π r l = 2 π θ
得到:
θ = l r \theta = \frac{l}{r} θ = r l
因此扇形面积公式还可以是:
π r 2 ⋅ l 2 π r = 1 2 r l = 1 2 r 2 θ \pi r^2 \cdot \frac{l}{2\pi r} = \frac{1}{2}rl = \frac{1}{2} r^2 \theta π r 2 ⋅ 2 π r l = 2 1 r l = 2 1 r 2 θ
本文链接:https://cs.pynote.net/math/202109097/
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