微积分，相关公式列表

2019-09-09 13:02

学习笔记  数学  微积分 

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The Theorem of Pythagoras(毕达哥拉斯定理)

$a^{2}+b^{2}=c^{2}$ if $c$ is the length of the hypotenuse and $a$ and $b$ are the shorter side lengths of a right-angled triangle.

Magnitude or absolute value(量或绝对值)

$|x|=\left\{\begin{array}{ll}{x} & {\text { if } x \geq 0} \\ {-x} & {\text { if } x<0}\end{array}\right.$

which is the distance from $x$ to 0 on the real line

Distance between points on the real line(实线上点之间的距离)

$|x-y|$ is the distance between $x$ and $y$ on the real line.

Distance between points on the plane(平面上点之间的距离)

$\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$ is the distance between points $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ on the $x y$ -plane.

Degrees and radians(度和弧度)

The angle $1^{\circ}$ equals $\frac{\pi}{180}$ radians. The angle 1 radian equals $\left(\frac{180}{\pi}\right)^{\circ} .$

Common trig values(常见的三角值)

$\sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}$
$\sin 60^{\circ}=\cos 30^{\circ}=\frac{\sqrt{3}}{2}$
$\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}$
$\tan 30^{\circ}=\frac{1}{\sqrt{3}}$
$\tan 60^{\circ}=\sqrt{3}$
$\tan 45^{\circ}=1$
$\cos 0=1$
$\cos \pi=-1$
$\sin \frac{\pi}{2}=1$
$\sin \frac{-\pi}{2}=-1$

Equation of a line(直线方程)

The line with slope $m$ and $y$ -intercept $k$ has equation $y=m x+k$

Equation of a circle(圆的方程)

The circle with centre $(a, b)$ and radius $r$ has equation

$(x-a)^{2}+(y-b)^{2}=r^{2}$

Area and perimeter of a circle(圆的面积和周长)

$A=\pi r^{2}$ and $P=2 \pi r$

Volume and surface area of a sphere(球体的体积和表面积)

$V=\frac{4}{3} \pi r^{3}$ and $S=4 \pi r^{2}$

Quadratic formula(二次公式)

The quadratic equation $a x^{2}+b x+c=0,$ where $a \neq 0$ has solutions

$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$

Inverse function(反函数)

If $y=f(x)$ satisfies the horizontal line test then $f$ is invertible with inverse
$f^{-1},$ which undoes $f$ in the sense that $x=f^{-1}(y) .$

Fractional powers and $n$ th roots(分数幂和n次方根)

$x^{1 / 2}=\sqrt{x}, \quad x^{1 / n}=\sqrt[n]{x}, \quad x^{m / n}=\left(x^{1 / n}\right)^{m}$

Exponential laws(指数定律)

$\quad a^{-x}=\frac{1}{a^{x}}, \quad a^{x} a^{y}=a^{x+y}, \quad \frac{a^{x}}{a^{y}}=a^{x-y}, \quad\left(\left(a^{x}\right)^{z}\right)=a^{x z}, \quad(a b)^{x}=a^{x} b^{x}$

Logs and exponentials undo each other(对数和指数相互求解)

For all $x,$ all positive $A$ and all positive $a \neq 1$
$\quad \log _{a}\left(a^{x}\right)=x, \quad \ln e^{x}=x, \quad a^{\log _{a} A}=A, \quad e^{\ln A}=A$.

Logarithmic laws(对数定律)

For all positive $x$ and $y,$ for all positive $a \neq 1,$ and for all real $k,$

• $\log _{a}(x y)=\log _{a} x+\log _{a} y, \quad \log _{a}(x / y)=\log _{a} x-\log _{a} y, \quad \log _{a}\left(x^{k}\right)=k \log _{a} x$
• $\ln (x y)=\ln x+\ln y, \quad \ln (x / y)=\ln x-\ln y, \quad \ln \left(x^{k}\right)=k \ln x$
• $a^{x}=e^{x \ln a}, \quad \log _{a} x=\frac{\ln x}{\ln a}$

Exponential growth and decay(指数增长和衰减)

$y=f(x)=A e^{k x}$ where $A>0,$ and $k>0$ for exponential growth, and $k<0$ for exponential decay.

Logistic function(逻辑函数)

$y=f(x)=\frac{M}{1+K e^{-k x}}$ for constants $M, K, k>0$

Average rate of change(平均变化率)

For $y=f(x)$ defined on the interval $[a, b],$ where $a<b,$ the average rate of change is $\frac{f(b)-f(a)}{b-a}$

Derivative(导数)

$y^{\prime}=\frac{d y}{d x}=f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$

Rules of differentiation(微分法则)

$(u v)^{\prime}=u v^{\prime}+v u^{\prime} \quad(\text { Product Rule })$

$\left(\frac{u}{v}\right)^{\prime}=\frac{v u^{\prime}-u v^{\prime}}{v^{2}} \quad(\text { Quotient Rule })$

$\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x} \quad \text { (Chain Rule) }$

Trick of Gauss(高斯算法)

$\sum_{i=1}^{n} i=1+2+\ldots+n=\frac{n(n+1)}{2}$

Even and odd functions(偶函数和奇函数)

$f$ is even if $f(x)=f(-x)$ and odd if $f(-x)=-f(x) .$ If $f$ is odd then $\int_{-a}^{a} f(x) d x=0 .$ If $f$ is even then $\int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x$

Standard integrals(标准积分)

$\int x^{n} d x=\frac{x^{n+1}}{n+1}+C \quad(n \neq-1)$

$\int \frac{d x}{x}=\ln |x|+C$

$\int e^{x} d x=e^{x}+C$

$\int \sin x d x=-\cos x+C$

$\int \cos x d x=\sin x+C$

$\int \sec ^{2} x d x=\tan x+C$

Fundamental Theorem of Calculus(微积分基本定理)

If $y=f(x)$ is a continuous function defined on a closed interval $[a, b]$ and $F(x)$ is any antiderivative of $f(x)$ with respect to $x$ then

$\int_{a}^{b} f(x) d x=F(b)-F(a)$

Integration by substitution(换元积分法)

$\int\left(f(u) \frac{d u}{d x}\right) d x=\int f(u) d u$

$\int f(g(x)) g^{\prime}(x) d x=\int f(u) d u$

$\int_{a}^{b} f(g(x)) g^{\prime}(x) d x=\int_{g(a)}^{g(b)} f(u) d u$