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24. Vl Ana1LinA Taylorapproximation $3.35   Add to cart

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24. Vl Ana1LinA Taylorapproximation

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Dies ist eine vollständige Vorlesungsmitschrift zur 24. Vorlesung des Moduls Analysis I und lineare Algebra für Ingenieurswissenschaften. In dieser Vorlesung wird die Taylorapproximation eingeführt und erklärt.

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  • February 1, 2023
  • 5
  • 2022/2023
  • Class notes
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f IR diffbar x ER

y flx.lt fix x x

lineareApproximation
flxkfG.ltflxlk x.IT xl fG.ltf'CxIA x.l

Was hat mit f flx.FI
gemeinsam l fkk.TK
fAFTAHRlxIf'lx.kTilx
fAI Tlx
E Ein
Restfliedstrebt
Gesucht ist ein Polynomenten Grades
fürdasgilt schneller alslinear
gegen


ftx.IT x fho Thot f Tn lol f k Tüll

p x an Zack xo Ja x x t h auf x p a la


p x Zazt32aglx x.lt th.tn e an x x
p Z az
iii Y
3 a e.ir n.a.name n


p klag

p x O
fürk n


Bedingung für das nie Tayberpolynem k auf A g FÜ
TKto FEIG x.ttYIlx x.Ft tEk x




THE.SI lx x

, Definition Taylor
polynom

Sei D R
f u Mal diffbar x D Dann heißt

Takt EILE x x n tes Taylorpolnen von
f im Entwicklungspaket
Xo
7Taylor
formel
Sei IE IR ein Intervall tot I f I IR diffbarSeiT.KI EIYE lx x.tk
RuIx sei der Fehler derApproximation f TG d h RAI fl l Tk
ERestglied

also gilt exakt flxt T.dk Rak
Dann
gilt d h derFehlergeht schneller als unter
ftp n o
Ordnung gegen 0


Ist f Intel Mal diffbar dann gilt
Rk FÜßE xd für ein
zwischen x und


Bsp 1 fx Polynom n ten Grades f GEO Ratko ´ denk 4


R Z
f ETH


fer eine Funktion mit f G Eo Rukko GEHTA Rt
n ten Gerades
f ist ein Polynom
3 Gtx et _O f G E f G f Aff A
f o 1

TK EIYIK x.T E.EE x



EoE.xk exx E Ex E

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