The Theorem of Pythagoras(毕达哥拉斯定理)

a2+b2=c2a^{2}+b^{2}=c^{2} if cc is the length of the hypotenuse and aa and bb are the shorter side lengths of a right-angled triangle.

Magnitude or absolute value(量或绝对值)

x={x if x0x if x<0|x|=\left\{\begin{array}{ll}{x} & {\text { if } x \geq 0} \\ {-x} & {\text { if } x<0}\end{array}\right.

which is the distance from xx to 0 on the real line

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

xy|x-y| is the distance between xx and yy on the real line.

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

(x1x2)2+(y1y2)2\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}} is the distance between points P(x1,y1)P\left(x_{1}, y_{1}\right) and Q(x2,y2)Q\left(x_{2}, y_{2}\right) on the xyx y -plane.

Degrees and radians(度和弧度)

The angle 11^{\circ} equals π180\frac{\pi}{180} radians. The angle 1 radian equals (180π).\left(\frac{180}{\pi}\right)^{\circ} .

Common trig values(常见的三角值)

sin30=cos60=12\sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}
sin60=cos30=32\sin 60^{\circ}=\cos 30^{\circ}=\frac{\sqrt{3}}{2}
sin45=cos45=12\sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}
tan30=13\tan 30^{\circ}=\frac{1}{\sqrt{3}}
tan60=3\tan 60^{\circ}=\sqrt{3}
tan45=1\tan 45^{\circ}=1
cos0=1\cos 0=1
cosπ=1\cos \pi=-1
sinπ2=1\sin \frac{\pi}{2}=1
sinπ2=1\sin \frac{-\pi}{2}=-1

Equation of a line(直线方程)

The line with slope mm and yy -intercept kk has equation y=mx+ky=m x+k

Equation of a circle(圆的方程)

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


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

A=πr2A=\pi r^{2} and P=2πrP=2 \pi r

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

V=43πr3V=\frac{4}{3} \pi r^{3} and S=4πr2S=4 \pi r^{2}

Quadratic formula(二次公式)

The quadratic equation ax2+bx+c=0,a x^{2}+b x+c=0, where a0a \neq 0 has solutions

x=b±b24ac2ax=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}

Inverse function(反函数)

If y=f(x)y=f(x) satisfies the horizontal line test then ff is invertible with inverse
f1,f^{-1}, which undoes ff in the sense that x=f1(y).x=f^{-1}(y) .

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

x1/2=x,x1/n=xn,xm/n=(x1/n)mx^{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)

ax=1ax,axay=ax+y,axay=axy,((ax)z)=axz,(ab)x=axbx\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,x, all positive AA and all positive a1a \neq 1
loga(ax)=x,lnex=x,alogaA=A,elnA=A\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 xx and y,y, for all positive a1,a \neq 1, and for all real k,k,

  • loga(xy)=logax+logay,loga(x/y)=logaxlogay,loga(xk)=klogax\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(xy)=lnx+lny,ln(x/y)=lnxlny,ln(xk)=klnx\ln (x y)=\ln x+\ln y, \quad \ln (x / y)=\ln x-\ln y, \quad \ln \left(x^{k}\right)=k \ln x
  • ax=exlna,logax=lnxlnaa^{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)=Aekxy=f(x)=A e^{k x} where A>0,A>0, and k>0k>0 for exponential growth, and k<0k<0 for exponential decay.

Logistic function(逻辑函数)

y=f(x)=M1+Kekxy=f(x)=\frac{M}{1+K e^{-k x}} for constants M,K,k>0M, K, k>0

Average rate of change(平均变化率)

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


y=dydx=f(x)=limh0f(x+h)f(x)h=limΔx0ΔyΔxy^{\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)

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

(uv)=vuuvv2( Quotient Rule )\left(\frac{u}{v}\right)^{\prime}=\frac{v u^{\prime}-u v^{\prime}}{v^{2}} \quad(\text { Quotient Rule })

dydx=dydududx (Chain 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)

i=1ni=1+2++n=n(n+1)2\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)

ff is even if f(x)=f(x)f(x)=f(-x) and odd if f(x)=f(x).f(-x)=-f(x) . If ff is odd then aaf(x)dx=0.\int_{-a}^{a} f(x) d x=0 . If ff is even then aaf(x)dx=20af(x)dx\int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x

Standard integrals(标准积分)

xndx=xn+1n+1+C(n1)\int x^{n} d x=\frac{x^{n+1}}{n+1}+C \quad(n \neq-1)

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

exdx=ex+C\int e^{x} d x=e^{x}+C

sinxdx=cosx+C\int \sin x d x=-\cos x+C

cosxdx=sinx+C\int \cos x d x=\sin x+C

sec2xdx=tanx+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)y=f(x) is a continuous function defined on a closed interval [a,b][a, b] and F(x)F(x) is any antiderivative of f(x)f(x) with respect to xx then

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

Integration by substitution(换元积分法)

(f(u)dudx)dx=f(u)du\int\left(f(u) \frac{d u}{d x}\right) d x=\int f(u) d u

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

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