Fórmula cuadrática

Si ax^2+bx+c=0, entonces x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}

Geometría

Triángulo de base b y altura hÁrea =\frac{1}{2} bh
Círculo de radio rCircunferencia =2 \pi rÁrea =\pi r^2
Esfera de radio rSuperficie = 4\pi r^2Volumen =\frac{4}{3} \pi r^3
Cilindro de radio r y altura hSuperficie curva =2 \pi r hVolumen =\pi r^2 h

Trigonometría

Identidades trigonométricas

  1. \sen \theta = 1/csc \ \theta
  2. \cos \theta = 1/sec \ \theta
  3. \tan \theta = 1/cot \ \theta
  4. \sen\ (90^0-\theta) = cos \ \theta
  5. \cos\ (90^0-\theta) = \sen \ \theta
  6. \tan \ (90^0-\theta) = \cot \ \theta
  7. \sen^2 \theta+\cos^2\theta=1
  8. \sec^2 \theta-\tan^2\theta=1
  9. \tan \theta=\sen \theta / \cos \theta
  10. \sen (\alpha \pm \beta)= \sen \alpha \cos \beta \pm \cos \alpha \sen \beta
  11. \cos (\alpha \pm \beta)= \cos \alpha \cos \beta \mp\sen \alpha \sen \beta
  12. \tan (\alpha \pm \beta)=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}
  13. \sen 2\theta=2 \sen \theta \cos \theta
  14. \cos 2\theta= \cos^2 \theta - \sen^2 \theta =2\cos^2 \theta -1=1-2 \sen^2 \theta
  15. \sen \alpha + \sen \beta=2 \sen \frac{1}{2} (\alpha+\beta) \cos \frac{1}{2} (\alpha - \beta)
  16. \cos \alpha + \cos \beta=2 \cos \frac{1}{2} (\alpha+\beta) \cos \frac{1}{2} (\alpha - \beta)

Triángulos

  1. Ley de los senos: \frac{a}{\sen \alpha}=\frac{b}{\sen \beta}=\frac{c}{\sen \gamma}
  2. Ley de los cosenos: c^2=a^2+b^2-2ab\cos \gamma
  1. Teorema de Pitágoras: a^2 + b^2 = c^2

Aplicaciones de la serie

  1. Teorema del binomio: (a+b)^n=a^n+na^{n-1}b+\frac{n(n-1)a^{n-2}b^2}{2!}+\frac{n(n-1)(n-2)a^{n-3}b^3}{3!}+...
  2. (1 \pm x)^n=1 \pm \frac{nx}{1!}+\frac{n(n-1)x^2}{2!} \pm ... \ (x^2<1)
  3. (1 \pm x)^{-n}=1 \mp \frac{nx}{1!}+\frac{n(n+1)x^2}{2!} \mp ... \ (x^2<1)
  4. \sen x=x- \frac{x^3}{3!}+\frac{x^5}{5!}- ...
  5. \cos x=1- \frac{x^2}{2!}+\frac{x^4}{4!}- ...
  6. \tan x=x+ \frac{x^3}{3}+\frac{2x^5}{15}+ ...
  7. e^x=1+x+\frac{x^2}{2!}+ ...
  8. ln (1+ x)=x- \frac{1}{2}x^2+\frac{1}{3}x^3- ... \ (|x|<1)

Derivadas

  1. \frac{d}{dx}[af(x)]=a\frac{d}{dx}f(x)
  2. \frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)
  3. \frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x)
  4. \frac{d}{dx}f(u)=[\frac{d}{du}f(u)]\frac{du}{dx}
  5. \frac{d}{dx}x^m=mx^{m-1}
  6. \frac{d}{dx} \sen x=\cos x
  7. \frac{d}{dx} \cos x=-\sen x
  8. \frac{d}{dx} \tan x=\sec^2 x
  9. \frac{d}{dx} \cot x=-\csc^2 x
  10. \frac{d}{dx} \sec x=\tan x \sec x
  11. \frac{d}{dx} \csc x=- \cot x \csc x
  12. \frac{d}{dx} e^x=e^x
  13. \frac{d}{dx} \ln x=\frac{1}{x}
  14. \frac{d}{dx} \sen^{-1} x=\frac{1}{\sqrt{1-x^2}}
  15. \frac{d}{dx} \cos^{-1} x=-\frac{1}{\sqrt{1-x^2}}
  16. \frac{d}{dx} \tan^{-1} x=\frac{1}{1+x^2}

Integrales

  1. \int af(x)dx=a \int f(x)dx
  2. \int [f(x)+g(x)]dx=\int f(x)dx+ \int g(x)dx
  3. \int x^mdx=\frac{x^{m+1}}{m+1} \ (m \neq -1) = \ln x \ (m=-1)
  4. \int \sen x dx=- \cos x
  5. \int \cos x dx= \sen x
  6. \int \tan x dx= \ln |sec \ x|
  7. \int \sen^2 ax dx=\frac{x}{2}- \frac{ \sen 2ax}{4a}
  8. \int \cos^2 ax dx=\frac{x}{2}+ \frac{ \sen 2ax}{4a}
  9. \int \sen ax \cos ax dx=-\frac{cos2ax}{4a}
  10. \int e^{ax} dx=\frac{1}{a} e^{ax}
  11. \int x e^{ax} dx= \frac{e^{ax}}{a^2}(ax-1)
  12. \int \ln ax dx = x \ln ax -x
  13. \int \frac{dx}{a^2+x^2} = \frac{1}{a} \tan^{-1} \frac{x}{a}
  14. \int \frac{dx}{a^2-x^2} = \frac{1}{2a} \ln |\frac{x+a}{x-a}|
  15. \int \frac{dx}{\sqrt{a^2+x^2}}=\senh^{-1} \frac{x}{a}
  16. \int \frac{dx}{\sqrt{a^2-x^2}}=\sen^{-1} \frac{x}{a}
  17. \int \sqrt{a^2+x^2}dx=\frac{x}{2} \sqrt{a^2+x^2}+\frac{a^2}{2} \senh^{-1} \frac{x}{a}
  18. \int \sqrt{a^2-x^2}dx=\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sen^{-1} \frac{x}{a}

\begin{itemize} \item Edad de Piedra \item Edad del Cobre \item Edad del Bronce \item Edad del Hierro \end{itemize}

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