From 1d2a91c67457f61e6fdc07a2914cf90b3d4679ad Mon Sep 17 00:00:00 2001 From: Erik Grobecker Date: Sat, 21 Sep 2024 16:36:25 +0200 Subject: [PATCH] =?UTF-8?q?habe=20"Untersuchungen=20von=20ganzrationalen?= =?UTF-8?q?=20Funktionen"=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit und ein bisschen qol --- klausurx.typ | 57 +++++++++++++++++++++++++++++++++++++++++----- pdfs/klausurx.pdf | Bin 187094 -> 195681 bytes 2 files changed, 51 insertions(+), 6 deletions(-) diff --git a/klausurx.typ b/klausurx.typ index 9ab5ca3..e43c9d7 100644 --- a/klausurx.typ +++ b/klausurx.typ @@ -26,24 +26,67 @@ - [Ausklammern] - Strecken, Stauchen, Verschieben und Drehen von Parabeln (für nen kleinen Teil der Klausur +#pagebreak() + = Untersuchungen von ganzrationalen Funktionen +werden auch Polynomfunktionen genannt + +=== Ableitungen + +Ableitungen erfolgen wiefolgt: + +$ +f(x)&=2x^2 \ +f'(x)&=2 dot 2 + x^(2-1) \ +&=4x \ +f''(x)&=4 dot 1 + x^(1-1) \ +f''(x)&=4 +$ + +=== Extremwerte + +H.p. = Hochpunkt\ +T.p. = Tiefpunkt + + +$ +text(H.p.)&: f'(x_(0))=0 text("und" ) f''(x_(0))<0 \ +text(T.p.)&: f'(x_(0))=0 text( "und" ) f''(x_(0))>0 +$ + +Also muss folgendes gemacht werden: + +$ +f(x)&=x^2 \ +f'(x)&=2x \ +f''(x)&=2&&|>0 \ \ +0&=2x&&|:2 \ +0&=x \ + \ +0& eq.not 2 &>0 \ +$ +Das heißt am y-Wert $0$ existiert ein Extremwert der noch berechnet werden müsste indem wir das Ergebniss der _ersten Ableitung_ wieder in die Funktion einsetzten, und es handelt sich um einen *Tiefpunkt* da der Wert größer als $0$ ist + +$ +f(0)=0^2=0 +$ + +Der Tiefpunkt ist also am Punkt (0|0). + +#pagebreak() = Funktionen mit Parametern (Funktionsscharen) == Untersuchungen === Nullstellen -Diese müssen für alle mögliche Werte von $a$ gelten. - -==== Beispiel - $ f_a (x)=x^2-2a x +8a -16 $ a) Zeigen sie, dass alle Graphen durch den Punkt S(4|0) verlaufen -#text(style: "italic", [Wir setzen also $4$ für $x$ ein]): +_Wir setzen also $4$ für $x$ ein_: $ f_a (4)&=4^2 -2a dot 4 +8a -16\ &=16-8a +8a-16\ @@ -52,6 +95,8 @@ f_a (4)&=4^2 -2a dot 4 +8a -16\ $ Das heißt das wir das $x$ des Punktes einsetzen mussten und das $y$ des Punktes als Ergbeniss erhalten mussten! +=== Extremwerte + b) Berechnen Sie die Koordinaten des Tiefpunktes der Graphen von $f_a$ in Abhängigkeit von $a$. $ @@ -62,7 +107,7 @@ f''_a (x)&=2\ \ 0&=x-a &&|-a\ a&=x $ -Da $f''(a)=2>0$ ist handelt es sich um einen #text(weight: "bold", [Tiefpunkt]).\ +Da $f''(a)=2>0$ ist handelt es sich um einen *Tiefpunkt*).\ Es gilt also#footnote([Hier wurde $x$ durch $a$ ersetzt]): $ f(a)&=a^2-2a^2+8a-16\ diff --git a/pdfs/klausurx.pdf b/pdfs/klausurx.pdf index d7653886ac9cd5c6eaa573b45429370390a7dbbb..7bd370e152978efcab0dd86839ac0faeb3304864 100644 GIT binary patch delta 127139 zcmY(rWl)|y&^BD4Kyfeb?(SCHwYa;x`&pp4J6yQC6u2i?Kw>8K$fG5($Ya20iAX7HN+&4t zLGUD@`eQ<~kg$-bCZ$7DBq{piBy6DkKZP3~2;L+I1%ae;2mpzVgq@9qlY^IuhlPW? zfeJDJDybX>Ie`d1sRJ74|5_xiLU(;*{h!s~g@h83NT4{mxl5lsAt52TS<+s-A&HW9 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