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Werktuigbouw formules: Calculus formuleblad

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Calculus formuleblad

Standaardlimieten voor functies

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{lim}under{x right 0} ~~{sin (x)}/x = 1

${lim}under{x right 0^+}~ x^p = 0$ (voor alle p > 0)

${lim}under{x right infty}~~ {x^p}/{a^x} = 0$ (voor alle p in bbR , a > 1)

${lim}under{x right infty}~~ {ln(x)}/{x^p} = 0$ (voor alle p > 0)

${lim}under{x right 0^+}~ x^p ln(x) = 0$ (voor alle p > 0)

${lim}under{x right infty}~~ (1 + alpha / x)^x = e^a$ (voor alle alpha in bbR )

Standaardlimieten voor rijen

${lim}under{n right infty} ~1/n^p = 0$ (als p > 0)

${lim}under{n right infty} ~r^n = 0$ (als delim{|}{r}{|} < 1)

{lim}under{n right infty} ~root{n}{n} = 1

{lim}under{n right infty} ~root{n}{alpha} = 1 (voor alle alpha > 0)

${lim}under{n right infty} ~(1 + alpha/n)^n = e^alpha$ (voor alle alpha in bbR )

Afgeleiden van standaardfuncties

(c)prime = 0

$(x^n)prime = n x^{n-1}$ (n in bbN)

$(x^{-n})prime = -n x^{-(n+1)}$ (n in bbN, x <> 0)

$(x^alpha)prime = alpha x^{alpha-1}$ (alpha in bbR, x > 0)

(e^x)prime = e^x

(ln x)prime = 1/x (x > 0)

(sin x)prime = cos x

(cos x)prime = - sin x

$(tan x)prime = 1/{cos^2 x} = 1 + tan^2 x$

$(arctan x)prime = 1/{1 + x^2}$

Standaard Taylor ontwikkelingen

$e^x = 1 + x + x^2/{2!} + x^3/{3!} + cdots + x^n/{n!} + R_n(x)$

$sin(x) = x - x^3/{3!} + x^5/{5!} - cdots + (-1)^n x^{2n+1)/{(2n+1)!}+ R_{2n+1}(x)$

$cos(x) = 1 - x^2/{2!} + x^4/{4!} - cdots + (-1)^n x^{2n)/{(2n)!}+ R_{2n}(x)$

$ln(1+x) = x - x^2/2 + x^3/3 - cdots + (-1)^{n-1} x^n/n+ R_n(x)$

$(1+x)^alpha = 1 + alpha x + (matrix{2}{1}{alpha 2}) x^2 + cdots + (matrix{2}{1}{alpha n}) x^n + R_n(x)$
Hierin is alpha in bbR en (matrix{2}{1}{alpha k}) = {alpha(alpha - 1) cdots (alpha - k + 1)}/{k!} .

$1/{1-x} = 1 + x + x^2 + x^3 + cdots + x^n + R_n(x)$

$arctan x = x - x^3/3 + x^5/5 - cdots + (-1)^n x^{2n+1)/{2n+1}+ R_{2n+1}(x)$

Met restterm $delim{|}{R_n(x)}{|} <= M/{(n+1)!} x^{n+1}$ waarbij $delim{|}{f^(n+1) (t)}{|} <= M$ op het interval delim{[}{0,x}{]} .

Lijst van Primitieven

$int{}{}{x^alpha dx} = 1/{alpha + 1} x^{alpha + 1} + c$ ( alpha in bbR, alpha <> -1 )

int{}{}{1/x dx} = ln delim{|}{x}{|} + c (x > 0 of x < 0)

$int{}{}{e^x dx} = e^x + c$

$int{}{}{a^x dx} = 1/{ln(a)} a^x + c$ (a > 0, a 1)

int{}{}{ln(x) dx} = x ln(x) - x + c (x > 0)

$int{}{}{1/{x^2 + 1} dx} = arctan(x) + c$

$int{}{}{1/{sqrt{1-x^2}} dx} = arcsin(x) + c$ (-1 < x < 1)

$int{}{}{{-1}/{sqrt{1-x^2}} dx} = arccos(x) + c$ (-1 < x < 1)

$int{}{}{1/{cos^2(x)} dx} = tan(x) + c$ ( x <> pi/2 + k pi, k in bbZ )

int{}{}{sin(x) dx} = - cos(x) + c

int{}{}{cos(x) dx} = sin(x) + c

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