142 lines
4.4 KiB
Typst
142 lines
4.4 KiB
Typst
#import "@preview/biceps:0.0.1": *
|
|
#import "@preview/cetz:0.4.2"
|
|
|
|
#import "lib/styles.typ": *
|
|
#import "lib/common.typ": *
|
|
|
|
#show: stdTemplate
|
|
#flexwrap( // Trigonometric formulas
|
|
main-spacing: 1mm,
|
|
cross-spacing: 1mm,
|
|
stdBlock([
|
|
== #hlHeading([Trig Identitäten])
|
|
$sin(x+y) = cos(x)sin(y) + sin(x)cos(y)$ \
|
|
$cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$ \
|
|
|
|
$cos(2x) = cos^2(x) - sin^2(x)$ \
|
|
$sin(2x) = 2sin(x)cos(x)$
|
|
|
|
#grid(
|
|
gutter: 5mm,
|
|
columns: (auto, auto),
|
|
[$cos^2(x) = (1 + cos(2x))/2$],
|
|
[$sin^2(x) = (1 - cos(2x))/2$]
|
|
)
|
|
|
|
$cos^2(x) + sin^2(x) = 1$
|
|
|
|
#grid(
|
|
gutter: 5mm,
|
|
columns: (auto, auto),
|
|
[$cos(-x) = cos(x)$],
|
|
[$sin(-x) = -sin(x)$],
|
|
)
|
|
|
|
Subsitution mit Hilfsvariable
|
|
|
|
#grid(
|
|
gutter: 5mm,
|
|
row-gutter: 3mm,
|
|
columns: (auto, auto),
|
|
[$tan(x)=sin(x)/cos(x)$],
|
|
[$cot(x)=cos(x)/sin(x)$],
|
|
[$tan(x)=-cot(x + pi/2)$],
|
|
[$cot(x)=-tan(x + pi/2)$],
|
|
[$cos(x - pi/2) = sin(x)$],
|
|
[$sin(x + pi/2) = cos(x)$],
|
|
)
|
|
$sin(x)cos(y) = 1/2sin(x - y) + 1/2sin(x + y)$
|
|
|
|
Für $x in [-1, 1]$ \
|
|
$arcsin(x) = -arccos(x) - pi/2 in [-pi/2, pi/2]$ \
|
|
$arccos(x) = -arcsin(x) + pi/2 in [0, pi]$
|
|
]),
|
|
sinTable,
|
|
stdBlock([
|
|
#grid(
|
|
columns:(auto, auto),
|
|
gutter: 1mm,
|
|
[
|
|
== #hlHeading([Folgen])
|
|
$ lim_(x->infinity) a_n $
|
|
- *Beschränkt*: $exists k in RR$ so dass $abs(a_n) <= k$
|
|
- $epsilon$-Interval: $x in (a - epsilon, a + epsilon) <=> abs(x - a) < epsilon$
|
|
- *Beweiß:* Induktion/Ungleichung
|
|
- Hat min. eine konvergent Teilfolge
|
|
- *Konvergent*:
|
|
- Es gibt $forall epsilon > 0$ eine Index $n_epsilon in NN$ sodass \ $abs(a_n - a) < epsilon space forall n > n_epsilon$
|
|
- Divergent $-> infinity$, wenn $forall k in RR : exists space a_n > k$
|
|
- Divergent $-> -infinity$, wenn $forall k in RR : exists space a_n < k$
|
|
- Genzwert is eindeutig
|
|
- *Monoton: steigen/fallend* $a_(n+1) gt.eq.lt a_n$
|
|
- *Beweisen:* Induktion mit \ $a_(n+1) gt.eq.lt a_n$ oder $a_(n+1) / a_(n) gt.lt 1 $
|
|
- *Konvergenz $a_n -> a$ $<=>$ beschränkt UND monoton*
|
|
- $<=>$ Alle Teilefolgen konvergent zu $a$
|
|
- Wenn Häufungspunk $eq.not$ $=>$ divergent
|
|
- Sandwitch-Theorem
|
|
|
|
=== Kriterien
|
|
$not$ Kriterium $=>$ $not$ Konvergenz *ABER*\ Kriterium $arrow.r.double.not$ Konvergenz
|
|
- Canchy-Kriterium: $forall space epsilon > 0 space exists space n,m > n_epsilon $ \ sodass $(a_n - a_m) < epsilon$
|
|
],
|
|
grid.vline(stroke: 0.1mm + black, position: start),
|
|
pad([
|
|
=== Grenzwert Finden:
|
|
- "Bottom up" von Bekannten Ausdrücken
|
|
- Fixpunk Gleösenichung l $a = f(a)$ für $f(a_n)$
|
|
- Bernoulli-Ungleichung für $(a_n)^n$ \
|
|
$(1 + a)^n >= 1 + n a$ für $a >= -1$
|
|
|
|
Für Konvergent Folgen:
|
|
#grid(
|
|
columns: (auto, auto),
|
|
align: bottom,
|
|
gutter: 2mm,
|
|
[$ lim_(n->infinity) (a_n + b_n) = a + b $],
|
|
grid.cell(
|
|
rowspan: 2,
|
|
[$ lim_(n->infinity) (a_n / b_n) = a / b $],
|
|
),
|
|
MathAlignLeft($ lim_(n->infinity) (a_n dot b_n) = a dot b $),
|
|
MathAlignLeft($ lim_(n->infinity) sqrt(a_n) = sqrt(a) $),
|
|
MathAlignLeft($ lim_(n->infinity) abs(a_n) = abs(a) $),
|
|
MathAlignLeft($ lim_(n->infinity) c dot a_n = c dot lim_(n->infinity) a_n $),
|
|
)
|
|
|
|
== Spezifische Folgen
|
|
#grid(
|
|
columns: (auto, auto, auto),
|
|
column-gutter: 4mm,
|
|
row-gutter: 2mm,
|
|
align: bottom,
|
|
MathAlignLeft($ lim_(n->infinity) 1/n = 0 $),
|
|
MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
|
|
MathAlignLeft($ lim_(n->infinity) q^n = 0 $),
|
|
grid.cell(colspan: 2, MathAlignLeft($ lim_(n->infinity) k = k, k in RR $)), [],
|
|
grid.cell(colspan: 2, MathAlignLeft($ exp(x) = e^x = lim_(n->infinity) (1 + x/n)^n $))
|
|
)
|
|
], left: 1mm)
|
|
)
|
|
|
|
|
|
|
|
]),
|
|
stdBlock([
|
|
== #hlHeading([Reihen])
|
|
=== Spezifische Reihen
|
|
#grid(columns: (auto, auto), column-gutter: 4mm, row-gutter: 2mm,
|
|
[
|
|
Geometrische Reihe:
|
|
$ sum_(n=0)^infinity $
|
|
- $ a_(n+q) = q a_n $
|
|
- Beschränkt: $abs(q) <= 1$
|
|
- Unbeschränkt: $abs(q) > 1$
|
|
],
|
|
[
|
|
Harmonische Reihe:
|
|
$ sum_(n=0)^infinity 1/n = +infinity $
|
|
]
|
|
)
|
|
]),
|
|
)
|