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alexander
9043
src/Analysis_rewrite.pdf
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9043
src/Analysis_rewrite.pdf
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File diff suppressed because one or more lines are too long
@@ -48,7 +48,7 @@
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$abs(x dot y) <= abs(abs(x) dot abs(y))$
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],
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[Geometrische Summenformel], [
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#MathAlignLeft($ sum_(k=1)^(n) k = (n(n+1))/2 $)
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#MathAlignLeft($ limits(sum)_(k=1)^(n) k = (n(n+1))/2 $)
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],
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[Bernoulli-Ungleichung ], [
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$(1 + a)^n x in RR >= 1 + n a$
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@@ -64,10 +64,58 @@
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[Gausklammer], [
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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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[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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]
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#bgBlock(fill: colorAllgemein)[
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#subHeading(fill: colorAllgemein)[Trigonmetrie]
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$sin(x+y) = cos(x)sin(y) + sin(x)cos(y)$ \
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$cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \
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$cos(2x) = cos^2(x) - sin^2(x)$ \
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$sin(2x) = 2sin(x)cos(x)$
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#grid(
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gutter: 5mm,
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columns: (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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)
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$cos^2(x) + sin^2(x) = 1$
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#grid(
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gutter: 5mm,
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columns: (auto, auto),
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[$cos(-x) = cos(x)$],
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[$sin(-x) = -sin(x)$],
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)
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Subsitution mit Hilfsvariable
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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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)
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$sin(x)cos(y) = 1/2sin(x - y) + 1/2sin(x + y)$
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Für $x in [-1, 1]$ \
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$arcsin(x) = -arccos(x) - pi/2 in [-pi/2, pi/2]$ \
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$arccos(x) = -arcsin(x) + pi/2 in [0, pi]$
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]
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#bgBlock(fill: colorFolgen)[
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#subHeading(fill: colorFolgen)[Folgen]
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$ lim_(x -> infinity) a_n $
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@@ -170,16 +218,82 @@
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#bgBlock(fill: colorReihen)[
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#subHeading(fill: colorReihen)[Reihen]
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$limits(lim)_(n->infinity) a_n != 0 => limits(sum)_(n=1)^infinity a_n$ konverigiert NICHT \
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- *Absolute Konvergenz* \
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$limits(sum)_(n=1)^infinity abs(a_n) = a => limits(sum)_(n=1)^infinity a_n$ konvergent
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- *Partialsummen* \
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ALLE Partialsummen von $limits(sum)_(k=1)^infinity abs(a)$ beschränkt\
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$=>$ _Absolute Konvergent_
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- *(Cauchy-Kriterium)*\
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konvergent wenn $forall epsilon > 0 space exists n_epsilon in NN$ \
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sodass $abs(s_n - s_m) = abs(limits(sum)_(k=m+1)^(n)) < epsilon space$ \
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$forall n_epsilon < m < n $
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- *Leibnitzkriterium* \
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Alternierend + Nullfolge \
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$=> limits(sum)_(n=1)^infinity (-1)^n dot a_n$ konvergent
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- *Vergleichskriterium* \
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$a_n, b_n : abs(a_n) <= b_n space forall n in NN > N_0, N_0 in NN$
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1. $limits(sum)_(n=0)^infinity b_n$ konvergent $=> limits(sum)_(n=0)^infinity abs(a_n)$ konvergent \
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Suche $b_n$ für Konvergenz
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2. $limits(sum)_(n=0)^infinity abs(a_n)$ divergent $=> limits(sum)_(n=0)^infinity b_n$ divergent \
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Suche $abs(a_n)$ für Divergenz
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Nützlich:
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- Dreiecksungleichung
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- $forall space n > N_0 in NN space exists k,q in RR$ \
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sodass $q > 1$: $n^k <= q^n$ (Potenz stärker Polynom)
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- *Quotientenkriterium und Wurzelkriterium*
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1. $rho = lim_(n -> infinity) abs((a_(n+1))/(a_n)) $
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2. $rho = lim_(n -> infinity) root(n, abs(a_(n+1))) $ \
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divergent: $rho > 1$, keine Aussage $rho = 1$, konvergent $rho < 1$
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- *Geometrische Reihe*
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$limits(sum)_(n=0)^infinity q^n$
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- konvergent $abs(q) < 1$, divergent $abs(q) >= 1$
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- Grenzwert: (Muss $n=0$) $=1/(1-q)$
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- *Harmonische Reihe* $limits(sum)_(n=0)^infinity 1/n = +infinity$
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- *Reihendarstellungen*
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1. $e^x = limits(sum)_(n=0)^infinity (x^n)/(n!)$
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2. $ln(x) = limits(sum)_(n=0)^infinity (-1)^n x^(n+1)$
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3. $sin(x) = limits(sum)_(n=0)^infinity $
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4. $cos(x) = limits(sum)_(n=0)^infinity $
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]
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#bgBlock(fill: colorReihen)[
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#subHeading(fill: colorReihen)[Potenzreihen]
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]
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#colbreak()
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#bgBlock(fill: colorAbleitung)[
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#subHeading(fill: colorAbleitung)[Funktionen]
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Sei $f : [a,b] -> RR$, stetig auf $x in [a,b]$
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- *Zwischenwertsatz* \
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$=> forall y in [f(a), f(b)] exists text("min. ein") x in [a,b] : f(x) = y$ \
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_Beweiß für mindest. n Nst_
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- *Satze von Rolle* \
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diffbar $x in (a,b)$\
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$f(a) = f(b) => exists text("min. ein") x_0 in (a,b) : f'(x_0) = 0$
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_Beweiß für max. n Nst, durchWiederspruchsbweiß mit $f(a)=f(b)=0$ und Wiederholte Ableitung_
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- *Mittelwertsatz*
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diffbar $x in (a,b)$ \
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$=> exists x_0 : f'(x_0)=(f(b) - f(a))/(a-b)$
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- *Monotonie* \
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$x in I : f'(x) < 0$: Streng monoton steigended \
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$x_0,x_1 in I, x_0 < x_1 => f(x_0) < f(x_1)$ \
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(Analog bei (streng ) steigned/fallended)
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]
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#bgBlock(fill: colorAbleitung)[
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@@ -256,7 +370,47 @@
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[$e^(x) : e^(x)$],
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)
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- Kettenregel: $f(g(x)) : f'(g(x)) dot g'(x)$
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]
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],
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#block([
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#set text(size: 10pt)
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#table(
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align: horizon,
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columns: (1fr, 1fr, 1fr),
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table.header([*$F(x)$*], [*$f(x)$*], [*$f'(x)$*]),
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row-gutter: 1mm,
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fill: (x, y) => if x == 0 { color.hsl(180deg, 89.47%, 88.82%) }
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else if x == 1 { color.hsl(180deg, 100%, 93.14%) } else
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{ color.hsl(180deg, 81.82%, 95.69%) },
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[$1/(q + x) x^(q+1)$], [$x^q$], [$q x^(q-1)$],
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[$ln abs(x)$], [$1/x$], [$-1/x^2$],
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[$x ln(a x) - x$], [$ln(a x)$], [$1 / x$],
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[$2/3 sqrt(a x^3)$], [$sqrt(a x)$], [$a/(2 sqrt(a x))$],
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[$e^x$], [$e^x$], [$e^x$],
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[$a^x/ln(a)$], [$a^x$], [$a^x ln(a)$],
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[$x arcsin(x) + sqrt(1 - x^2)$],
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[$arcsin(x)$], [$1/sqrt(1 - x^2)$],
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[$x arccos(x) - sqrt(1 - x^2)$],
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[$arccos(x)$], [$-1/sqrt(1 - x^2)$],
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[$x arctan(x) - 1/2 ln abs(1 + x^2)$],
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[$arctan(x)$], [$1/(1 + x^2)$],
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[$x op("arccot")(x) + \ 1/2 ln abs(1 + x^2)$],
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[$op("arccot")(x)$], [$-1/(1 + x^2)$],
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[$x op("arsinH")(x) + \ sqrt(1 + x^2)$],
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[$op("arsinH")(x)$], [$1/sqrt(1 + x^2)$],
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[$x op("arcosH")(x) + \ sqrt(1 + x^2)$],
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[$op("arcosH")(x)$], [$1/sqrt(x^2-1)$],
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[$x op("artanH")(x) + \ 1/2 ln(1 - x^2)$],
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[$op("artanH")(x)$], [$1/(1 - x^2)$],
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)
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])
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#colbreak()
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]
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