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\topmatter
\title Formelsamling MA 100 \endtitle
%\rightheadtext{}
\endtopmatter

\define\N{\Bbb N}
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\document

\subhead Eksponentialfunksjonen \endsubhead

Derivasjon:
$$
D[e^x] = e^x
$$

Identiteter:
$$
e^{x+y} = e^x e^y \qquad \qquad
e^{-x} = {1 \over e^x} \qquad \qquad
(e^x)^y = e^{xy}
$$

\subhead Logaritmefunksjonen \endsubhead

Derivasjon:
$$
D[\ln|x|] = {1 \over x}
$$

Identiteter:
$$
\alignat 2
\ln(xy) &= \ln x + \ln y &\qquad \qquad
\ln\left({1\over x}\right) &= -\ln x \\
\ln\left({x \over y}\right) &= \ln x - \ln y &\qquad \qquad
\ln x^a &= a \ln x
\endalignat
$$
hvor $x, y > 0$.

\subhead Trigonometriske funksjoner \endsubhead

Derivasjon:
$$
\alignat 2
D[\sin x] &= \cos x &\qquad \qquad
D[\tan x] &= {1 \over \cos^2 x} \\
D[\cos x] &= -\sin x &\qquad \qquad
D[\cot x] &= -{1 \over \sin^2 x}
\endalignat
$$

Identiteter:
$$
\gather
\sin^2 x + \cos^2 x = 1 \\
\aligned
\sin(x+y) &= \sin x \cos y + \cos x \sin y \\
\cos(x+y) &= \cos x \cos y - \sin x \sin y \\
\sin 2x &= 2 \sin x \cos x \\
\cos 2x &= \cos^2 x - \sin^2 x \\
 &= 2 \cos^2 x - 1 \\
 &= 1 - 2 \sin^2 x
\endaligned \\
\alignedat 2
\sin x &= \pm {\tan x \over \sqrt{1 + \tan^2 x}} &\qquad \qquad 
\sin x &= {2 \tan {x\over2} \over 1 + \tan^2 {x\over2}} \\
\cos x &= \pm {1 \over \sqrt{1 + \tan^2 x}} &\qquad \qquad
\cos x &= {1 - \tan^2 {x\over2} \over 1 + \tan^2 {x\over2}}
\endalignedat
\endgather
$$

\subhead Arcusfunksjonene \endsubhead

Derivasjon:
$$
\align
D[\arcsin x] &= {1 \over \sqrt{1-x^2}} \\
D[\arccos x] &= - {1 \over \sqrt{1-x^2}} \\
D[\arctan x] &= {1 \over 1 + x^2}
\endalign
$$

\subhead Rekker \endsubhead

$$
\align
{1 \over 1-x} &= \sum_{n=0}^\infty x^n \,, \quad
\text{konvergensomr{\aa}de $(-1, 1)$ (geometrisk rekke)} \\
(1+x)^\alpha &= \sum_{n=0}^\infty \binom{\alpha}{n} x^n \,, \quad
\text{konvergensradius $1$ for $\alpha \notin \N$, (binomisk rekke)} \\
e^x &= \sum_{n=0}^\infty {x^n \over n!} \\
\sin x &= \sum_{n=0}^\infty (-1)^n {x^{2n+1} \over (2n+1)!} \\
\cos x &= \sum_{n=0}^\infty (-1)^n {x^{2n} \over (2n)!} \\
\ln(1-x) &= -\sum_{n=1}^\infty {x^n \over n} \,, \quad
\text{konvergensomr{\aa}de $[-1, 1)$} \\
\arctan x &= \sum_{n=0}^\infty (-1)^n {x^{2n+1} \over 2n+1} \,, \quad
\text{konvergensomr{\aa}de $[-1, 1]$}
\endalign
$$

\subhead Diverse \endsubhead

Binomialformelen:
$$
(a+b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k
$$
De Moivres formel:
$$
(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)
$$
Newtons metode:
$$
x_{n+1} = x_n - {f(x_n) \over f'(x_n)}
$$

\enddocument
