Added Color Boxes
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alexander
@@ -50,159 +50,177 @@
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#let colorAbleitung = color.hsl(356.92deg, 92.13%, 75.1%)
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#let colorAbleitung = color.hsl(356.92deg, 92.13%, 75.1%)
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#let colorIntegral = color.hsl(34.87deg, 92.13%, 75.1%)
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#let colorIntegral = color.hsl(34.87deg, 92.13%, 75.1%)
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#let bgBlock(body, fill: color) = block(body, fill:fill.lighten(80%), width: 100%, inset: (bottom: 2mm))
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#columns(4, gutter: 2mm)[
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#columns(4, gutter: 2mm)[
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#subHeading(fill: colorAllgemein, it: [Allgemeins])
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#bgBlock(fill: colorAllgemein)[
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#grid(
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#subHeading(fill: colorAllgemein, it: [Allgemeins])
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columns: (auto, auto),
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#grid(
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row-gutter: 2mm,
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columns: (auto, auto),
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column-gutter: 3mm,
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row-gutter: 2mm,
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[Dreiecksungleichung], [
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column-gutter: 3mm,
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$abs(x + y) <= abs(x) + abs(y)$ \
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[Dreiecksungleichung], [
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$abs(abs(x) - abs(y)) <= abs(x - y)$
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$abs(x + y) <= abs(x) + abs(y)$ \
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],
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$abs(abs(x) - abs(y)) <= abs(x - y)$
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[Cauchy-Schwarz-Ungleichung], [
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],
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$abs(x dot y) <= abs(abs(x) dot abs(y))$
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[Cauchy-Schwarz-Ungleichung], [
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],
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$abs(x dot y) <= abs(abs(x) dot abs(y))$
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[Geometrische Summenformel], [
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],
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#MathAlignLeft($ sum_(k=1)^(n) k = (n(n+1))/2 $)
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[Geometrische Summenformel], [
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],
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#MathAlignLeft($ sum_(k=1)^(n) k = (n(n+1))/2 $)
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[Bernoulli-Ungleichung ], [
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],
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[Bernoulli-Ungleichung ], [
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$(1 + a)^n >= 1 + n a$
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],
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[Binomialkoeffizient], [
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$binom(n, k) = (n!)/(k!(n-k)!)$
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],
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[Binomische Formel], [
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#MathAlignLeft($ (a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $)
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],
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[Fakultäten], [$ 0! = 1! = 1 $],
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)
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]
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen, it: [Folgen])
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$ lim_(x -> infinity) a_n $
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*Beschränkt:* $exists k in RR$ sodass $abs(a_n) <= k$
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- Beweiße: durch Induktion
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- Beweiße: Hat min. ein konvergent Teilefolge
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- (Beweiße: Ungleichung $abs(a_n) <= k$)
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*Monoton fallend/steigended*
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- Beweise: Induktion
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#grid(columns: (1fr, 1fr),
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gutter: 1mm,
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row-gutter: 2mm,
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align(top+center, [*Fallend*]), align(top+center, [*Steigend*]),
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[$ a_(n+1) <= a_(n) $],
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[$ a_(n+1) >= a_(n) $],
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[$ a_(n+1)/a_(n) < 1 $],
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[$ a_(n+1)/a_(n) > 1 $],
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)
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*Konvergentz Allgemein*
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$ lim_(n -> infinity) a_n = a $
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$forall epsilon > 0 space exists n_epsilon in NN$ sodass \
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- Konvergent $-> a$: $a_n in [a - epsilon, a + epsilon] $
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- Divergent $-> infinity$: $a_n in [epsilon, infinity) $
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- Divergent $-> infinity$: $a_n in (-infinity, epsilon) $
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$space forall n > n_epsilon$
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*Konvergentz Häufungspunkte*
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- $a_n -> a <=>$ Alle Teilfolgen $-> a$
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*Konvergenz Beweißen*
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- Monoton UND Beschränkt $=>$ Konvergenz
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NICHT Umgekehert
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- (Cauchyfolge \
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$forall epsilon > 0 space exists n_epsilon in NN space$ sodass \
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$forall m,n >= n_epsilon : abs(a_n - a_m) < epsilon$ \
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Cauchyfolge $=>$ Konvergenz)
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- $a_n$ unbeschränkt $=>$ divergenz
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*Konvergent Grenzwert finden*
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- Von Bekannten Ausdrücken aufbauen
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- Fixpunk Gleichung: $a = f(a)$ \
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für rekusive $a_(n+1) = f(a_n)$ (Zu erst machen!)
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- Bernoulli-Ungleichung Folgen der Art $(a_n)^n$: \
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$(1 + a)^n >= 1 + n a$
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$(1 + a)^n >= 1 + n a$
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],
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- Sandwitchtheorem:\
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[Binomialkoeffizient], [
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$b_n -> x$: $a_n <= b_n <= c_n$, wenn $a_n -> x$ und $c_n -> x$ \
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$binom(n, k) = (n!)/(k!(n-k)!)$
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$b_n -> -infinity$: $b_n <= c_n$, wenn $c_n -> -infinity$ \
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],
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$b_n -> +infinity$: $c_n <= b_n $, wenn $a_n -> +infinity$
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[Binomische Formel], [
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- Zwerlegen in Konvergente Teil folgen \
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#MathAlignLeft($ (a + b)^n = sum^(n)_(k=0) binom(n,k) a^(n-k) b^k $)
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(Vorallem bei $(-1)^n dot a_n$)
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],
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]
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[Fakultäten], [$ 0! = 1! = 1 $],
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)
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen, it: [Konvergent Folge Regeln])
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#grid(
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columns: (auto, auto),
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align: bottom,
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gutter: 2mm,
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[$ lim_(n->infinity) (a_n + b_n) = a + b $],
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grid.cell(
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rowspan: 2,
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[$ lim_(n->infinity) (a_n / b_n) = a / b $ für ($b != 0$)],
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),
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MathAlignLeft($ lim_(n->infinity) (a_n dot b_n) = a dot b $),
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MathAlignLeft($ lim_(n->infinity) sqrt(a_n) = sqrt(a) $),
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MathAlignLeft($ lim_(n->infinity) abs(a_n) = abs(a) $),
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MathAlignLeft($ lim_(n->infinity) c dot a_n = c dot lim_(n->infinity) a_n $),
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)
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]
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#subHeading(fill: colorFolgen, it: [Folgen])
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#bgBlock(fill: colorFolgen)[
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$ lim_(x -> infinity) a_n $
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#subHeading(fill: colorFolgen, it: [Bekannte Folgen])
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#grid(
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columns: (auto, auto, auto),
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column-gutter: 4mm,
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row-gutter: 2mm,
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align: bottom,
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MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
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MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
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MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
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grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $)), [],
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grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) k = k, k in RR $)), [],
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grid.cell(colspan: 2, MathAlignLeft($ exp(x) = e^x = lim_(n->infinity) (1 + x/n)^n $))
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)
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]
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*Beschränkt:* $exists k in RR$ sodass $abs(a_n) <= k$
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#bgBlock(fill: colorFolgen)[
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- Beweiße: durch Induktion
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#subHeading(fill: colorFolgen, it: [Teilfolgen])
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- Beweiße: Hat min. ein konvergent Teilefolge
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$ a_k subset a_n space (text("z.B") k= 2n + 1) $
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- (Beweiße: Ungleichung $abs(a_n) <= k$)
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- Index muss streng monoton steigen!
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- Beschränkte $a_n => text("min eine konvergente") a_k$
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- Konvergenz-Werte von $a_k$ sind Häufungspunkte
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- Wenn alle $a_k$ gegen #underline([genau eine]) Häufungspunk konverigiert $<=> a_n$ konvergent
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]
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*Monoton fallend/steigended*
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#bgBlock(fill: colorReihen)[
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- Beweise: Induktion
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#subHeading(fill: colorReihen, it: [Reihen])
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#grid(columns: (1fr, 1fr),
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]
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gutter: 1mm,
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row-gutter: 2mm,
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align(top+center, [*Fallend*]), align(top+center, [*Steigend*]),
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[$ a_(n+1) <= a_(n) $],
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[$ a_(n+1) >= a_(n) $],
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[$ a_(n+1)/a_(n) < 1 $],
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[$ a_(n+1)/a_(n) > 1 $],
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)
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*Konvergentz Allgemein*
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#bgBlock(fill: colorReihen)[
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$ lim_(n -> infinity) a_n = a $
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#subHeading(fill: colorReihen, it: [Potenzreihen])
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]
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$forall epsilon > 0 space exists n_epsilon in NN$ sodass \
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- Konvergent $-> a$: $a_n in [a - epsilon, a + epsilon] $
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- Divergent $-> infinity$: $a_n in [epsilon, infinity) $
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- Divergent $-> infinity$: $a_n in (-infinity, epsilon) $
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$space forall n > n_epsilon$
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*Konvergentz Häufungspunkte*
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- $a_n -> a <=>$ Alle Teilfolgen $-> a$
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*Konvergenz Beweißen*
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- Monoton UND Beschränkt $=>$ Konvergenz
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NICHT Umgekehert
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- (Cauchyfolge \
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$forall epsilon > 0 space exists n_epsilon in NN space$ sodass \
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$forall m,n >= n_epsilon : abs(a_n - a_m) < epsilon$ \
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Cauchyfolge $=>$ Konvergenz)
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- $a_n$ unbeschränkt $=>$ divergenz
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*Konvergent Grenzwert finden*
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- Von Bekannten Ausdrücken aufbauen
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- Fixpunk Gleichung: $a = f(a)$ \
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für rekusive $a_(n+1) = f(a_n)$ (Zu erst machen!)
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- Bernoulli-Ungleichung Folgen der Art $(a_n)^n$: \
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$(1 + a)^n >= 1 + n a$
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- Sandwitchtheorem:\
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$b_n -> x$: $a_n <= b_n <= c_n$, wenn $a_n -> x$ und $c_n -> x$ \
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$b_n -> -infinity$: $b_n <= c_n$, wenn $c_n -> -infinity$ \
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$b_n -> +infinity$: $c_n <= b_n $, wenn $a_n -> +infinity$
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- Zwerlegen in Konvergente Teil folgen \
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(Vorallem bei $(-1)^n dot a_n$)
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#subHeading(fill: colorFolgen, it: [Konvergent Folge Regeln])
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#grid(
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columns: (auto, auto),
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align: bottom,
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gutter: 2mm,
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[$ lim_(n->infinity) (a_n + b_n) = a + b $],
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grid.cell(
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rowspan: 2,
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[$ lim_(n->infinity) (a_n / b_n) = a / b $ für ($b != 0$)],
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),
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MathAlignLeft($ lim_(n->infinity) (a_n dot b_n) = a dot b $),
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MathAlignLeft($ lim_(n->infinity) sqrt(a_n) = sqrt(a) $),
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MathAlignLeft($ lim_(n->infinity) abs(a_n) = abs(a) $),
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MathAlignLeft($ lim_(n->infinity) c dot a_n = c dot lim_(n->infinity) a_n $),
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)
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#subHeading(fill: colorFolgen, it: [Bekannte Folgen])
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#grid(
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columns: (auto, auto, auto),
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column-gutter: 4mm,
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row-gutter: 2mm,
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align: bottom,
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MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
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MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
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MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
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grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) sqrt(n) = + infinity $)), [],
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grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) k = k, k in RR $)), [],
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grid.cell(colspan: 2, MathAlignLeft($ exp(x) = e^x = lim_(n->infinity) (1 + x/n)^n $))
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)
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#subHeading(fill: colorFolgen, it: [Teilfolgen])
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$ a_k subset a_n space (text("z.B") k= 2n + 1) $
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- Index muss streng monoton steigen!
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- Beschränkte $a_n => text("min eine konvergente") a_k$
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- Konvergenz-Werte von $a_k$ sind Häufungspunkte
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- Wenn alle $a_k$ gegen #underline([genau eine]) Häufungspunk konverigiert $<=> a_n$ konvergent
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#subHeading(fill: colorReihen, it: [Reihen])
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#subHeading(fill: colorReihen, it: [Potenzreihen])
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#colbreak()
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#colbreak()
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#subHeading(fill: colorAbleitung, it: [Funktionen])
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- $f(x)$ ist an der Stelle $x_0 in DD$ diffbar wenn \
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#MathAlignLeft($ f'(x_0) = lim_(x->x_0 plus.minus) (f(x_0 + h - f(x_0))/h) $)
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- $f(x)$ diffbar $=>$ $f(x)$ stetig
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#subHeading(fill: colorAbleitung, it: [Ableitung])
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#bgBlock(fill: colorAbleitung)[
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#grid(
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#subHeading(fill: colorAbleitung, it: [Funktionen])
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row-gutter: 3mm,
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- $f(x)$ ist an der Stelle $x_0 in DD$ diffbar wenn \
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columns: (1fr, 1fr),
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#MathAlignLeft($ f'(x_0) = lim_(x->x_0 plus.minus) (f(x_0 + h - f(x_0))/h) $)
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grid.cell(
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- $f(x)$ diffbar $=>$ $f(x)$ stetig
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colspan: 2,
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]
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[$f(x) + g(x) : f'(x) + g'(x) $]
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),
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#bgBlock(fill: colorAbleitung)[
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grid.cell(
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#subHeading(fill: colorAbleitung, it: [Ableitung])
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colspan: 2,
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#grid(
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[$f(x) dot g(x) : f'(x)g(x) + f(x)g'(x) $]
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row-gutter: 3mm,
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),
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columns: (1fr, 1fr),
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grid.cell(
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grid.cell(
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colspan: 2,
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colspan: 2,
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[#MathAlignLeft($ f(x)/g(x) : (f'(x)g(x) - f(x)g'(x))/(g(x)^2) $)]
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[$f(x) + g(x) : f'(x) + g'(x) $]
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),
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),
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[$f(x) = c : f'(x) = 0$],
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grid.cell(
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[$c dot f(x) : c dot f'(x)$],
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colspan: 2,
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[$(x^(-n)) n in NN : n x^(n-1)$],
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[$f(x) dot g(x) : f'(x)g(x) + f(x)g'(x) $]
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)
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),
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grid.cell(
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colspan: 2,
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[#MathAlignLeft($ f(x)/g(x) : (f'(x)g(x) - f(x)g'(x))/(g(x)^2) $)]
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),
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[$f(x) = c : f'(x) = 0$],
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[$c dot f(x) : c dot f'(x)$],
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[$(x^(-n)) n in NN : n x^(n-1)$],
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)
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]
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#colbreak()
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#colbreak()
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]
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]
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Reference in New Issue
Block a user