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alexander
@@ -76,30 +76,19 @@
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*Binomische Formel*\
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$(a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $ \
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],
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[
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*Bekannte Werte* \
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$e approx 2.71828$ ($2 < e < 3$) \
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$pi approx 3.14159$ ($3 < pi < 4$)
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],
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[
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*Gaußklammer*: \
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$floor(x) = text("floor")(x)$ \
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$ceil(x) = text("ceil")(x)$ \
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],
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[
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*Fakultäten* $0! = 1! = 1$ \
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],
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[
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*Mitternachtsformel*
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$x_(1,2) = (-b plus.minus sqrt(b^2 + 4a c))/(2a)$
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],
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[
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*Binomische Formel*\
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$(a + b)^2 = a^2 + 2a b + b^2$\
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$(a - b)^2 = a^2 - 2a b + b^2$\
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$(a + b)(a - b) = a^2 - b^2$\
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]
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)
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$bold("Fakultäten") 0! = 1! = 1\
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e approx 2.71828 quad quad quad pi approx 3.14159
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$
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]
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// Complex Zahlen
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@@ -144,22 +133,17 @@
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gutter: 2mm,
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columns: (auto, auto, auto),
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$cos^2(x) = (1 + cos(2x))/2$,
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$sin^2(x) = (1 - cos(2x))/2$,
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$cos(-x) = cos(x)$,
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$sin(-x) = -sin(x)$,
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grid.cell(colspan: 2, $cos^2(x) + sin^2(x) = 1$)
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$sin^2(x) = (1 - cos(2x))/2$
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)
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Subsitution mit Hilfsvariable
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$tan(x)=sin(x)/cos(x) = -cot(x + pi/2)$
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$cot(x)=cos(x)/sin(x)=-tan(x + pi/2)$
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#grid(
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gutter: 5mm,
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row-gutter: 3mm,
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columns: (auto, auto),
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[$tan(x)=sin(x)/cos(x)$],
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[$cot(x)=cos(x)/sin(x)$],
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[$tan(x)=-cot(x + pi/2)$],
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[$cot(x)=-tan(x + pi/2)$],
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[$cos(x - pi/2) = sin(x)$],
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[$sin(x + pi/2) = cos(x)$],
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@@ -377,13 +361,13 @@
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*Reihendarstellungen*
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#grid(
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columns: (1fr, 1fr),
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columns: (1fr),
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gutter: 3mm,
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row-gutter: 3mm,
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$e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$,
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$ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$,
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$sin(x) = limits(sum)_(n=0)^infinity (-1)^n (z^(2n+1))/((2n + 1)!)$,
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$cos(x) = limits(sum)_(n=0)^infinity (-1)^n (z^(2n))/((2n)!)$
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$e^x = limits(sum)_(n=0)^m (x^n)/(n!) + O(x^(m+1))$,
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$ln(x) = limits(sum)_(n=0)^m (-1)^n x^(n+1) + O(x^(m+1))$,
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$sin(x) = limits(sum)_(n=0)^m (-1)^n (z^(2n+1))/((2n + 1)!) + O(x^(2m + 3))$,
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$cos(x) = limits(sum)_(n=0)^m (-1)^n (z^(2n))/((2n)!) + O(x^(2m + 2))$
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)
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]
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@@ -446,7 +430,7 @@
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*Allgemein*
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$f(x)$ ist stetig wenn: \
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$ limits(lim)_(x->x_0-) f(x) = limits(lim)_(x->x_0+) f(x) = f(x_0) $ \
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$limits(lim)_(x->x_0-) f(x) = limits(lim)_(x->x_0+) f(x) = f(x_0)$ \
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$x in DD$ Beachten! Definitionslücken $!=$ unstätig \
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Definition gilt auch für $I subset RR$
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@@ -613,12 +597,25 @@
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*Subsitution*
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$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot g'(x) d x$
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$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) d x = integral_g(x_0)^g(x_1) f(t) dot <1/(g'(x)) d t$
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1. Ersetzung: $t := g(x)$
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2. Umformen:
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$(d y)/(d x) = g'(x)$
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3. $x$-kürzen sich weg
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$(d t)/(d x) = g'(x)$
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*Weierstrass Subsitution/Brechstange* \
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Subsitution: $t = tan(x/2)$
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#grid(
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columns: (1fr, 1fr),
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row-gutter: 2.8mm,
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$d x = 2/(1+t^2) d t$,
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$sin(x) = 2t / (1 + t^2)$,
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$tan(x) = (2t)/(1-t^2)$,
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$cos(x) = (1-t^2) / (1 + t^2)$,
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)
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])
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#bgBlock(fill: colorIntegral, [
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@@ -653,11 +650,10 @@
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*Subsitution*
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$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot 1/(g'(x)) d x$
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$integral_(x_0)^(x_1) f\(underbrace(g(x), "t")\) dot 1/(g'(x)) d t$
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1. Ersetzung: $ d x := d t dot g'(x)$ und $t := g(x)$
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1. Ersetzung: $ d x := 1/g'(x) dot d t$ und $t := g(x)$
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2. Grenzen: $t_0 = g(x_0)$, $t_1 = g(x_1)$
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3. $x$-kürzen sich weg
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*Absolute "Konvergenz"* \
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Wenn $g(x)$ konvergent,
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@@ -680,20 +676,13 @@
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5. $A,B,...$ :
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Nst einsetzen, dann Koeffizientenvergleich
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6. *Intergral wiederzusammen setzen $+c$*
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7. Summen teile Integrieren
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$delta = 4a - b^2$
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#grid(columns: (auto, auto),
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row-gutter: 2mm,
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column-gutter: 2mm,
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$integral 1/(x - x_0)$, $ln abs(x - x_0)$,
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$integral 1/((x - x_0)^n)$, $-1/((n-1)(x-x_0)^(n-1))$,
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$integral 1/(x^2 + b x + c)$, $2/sqrt(delta) arctan((2x + b)/sqrt(delta))$,
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$integral 1/((x^2 + b x + c)^n)$, $(2x + b)/((n-1)(sigma)(x^2+b x +c)^(n-1)) + \
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(2(2n-3))/((n-1)(delta)) + (C )
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$,
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)
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7. *Summen teile Integrieren*
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$integral 1/(x-a) d x = ln(x - a) + c\
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integral 1/(x-a)^n d x = - 1/(n-1) 1/(x - a)^(n-1) + c quad "für" n >= 2 \
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integral 1/((x - a)^2 + b^2) d x = 1/b arctan((x - a)/b) + c quad "für" n > 0\
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integral (x - a)/((x-a)^a + b^2) d x = 1/2 ln((x-a)^2 + b^2) + c
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$
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])
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@@ -703,6 +692,8 @@
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#sinTable
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])
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/*
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// Notwending und Hinreichend
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#bgBlock(fill: colorAllgemein, [
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#subHeading(fill: colorAllgemein)[Notwending und Hinreiched]
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@@ -718,4 +709,62 @@
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$not "hin." arrow.r.double.not "Satz"$,
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)
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])
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*/
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// Taylor Reihen
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#bgBlock(fill: colorAbleitung)[
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#subHeading(fill: colorAbleitung)[Taylorreihe]
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$T_m (x_0;x) = sum^m_(k=0) (f^((k))(x_0))/(k!) (x-x_0)^k$
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$f(x) = T_m (x_0;x) + R_(m+1)(x_0;x)$
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*Restglied* \
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$I = (a,b) quad x_0,x in I$
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$R_(m+1) (x) = 1/(m!) integral_(x_0)^x (x-t)^m f^((m+1))(t) d t$ \
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$forall x in I space space exists xi_x in I "sodass" \
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R_(m+1)(x_0;x) = (f^((m+1))(xi_x))/((m + 1)!) (x - x_0)^(m+1)\
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= f(x_0 + h) - T_m (x_0; x_0 + h) = o(h^m) = O(h^(m+1))$
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]
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// Lamdauer Notation
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#bgBlock(fill: colorFolgen, [
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#subHeading(fill: colorFolgen)[Landau Notation]
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$f(x) = o(g(x)) "wenn" lim_(x->a) f(x)/g(x) = 0 \
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f(x) = O(g(x)) "wenn" abs(f(x)) <= abs(g(g))$
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*Rechen Regelen* \
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- $f = o(g) => f = O(g)$
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- $f_1 + f_2 = O\/o(f_1) + O\/o(f_2)$
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- $f_1 dot f_2 = O\/o(f_1 dot f_2)$
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])
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// Kurven
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#bgBlock(fill: colorAbleitung)[
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#subHeading(fill: colorAbleitung)[Kurven]
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Länge einer Kurve $k(t)$: $L(k) = integral_a^b norm(k'(t)) d t$
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Umparametrisierung: $h(tau)$ *streng monoton steigended* \
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$h(tau): [a,b] -> [overline(a), overline(b)]$
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$overline(k): [overline(a), overline(b)] -> R^n, quad overline(k) = k(h(tau))$
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Parametriesierung nach Länge: \
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$s(t) = integral_a^t norm(k'(tau)) d tau quad overline(k)(tau) = k(s^(-1)(tau))$
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Wenn $k$ nach Länge param.: $T(t) =k'(t)$
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Tangentenvektor: $T(t) = (k'(t))/norm(k'(t))$ \
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Krümmung: $kappa(t) = 1/(s'(t)) norm(T'(t))$
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$RR^2: kappa(t) = abs(x'(t)y''(t) - y'(t)x''(t))/((x'(t)^2 + y'(r)^2)^(3/2))$ \
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Param Länge: $kappa(t) = abs(x'(t)y''(t) - y'(t)x''(t))$
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]
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]
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