Taylorpolynomier Funktion af flere variable
Taylorpolynomier Funktion af flere variable
Taylorpolynomier Funktion af flere variable
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Taylors formel med Lagrange’s restled<br />
<strong>Taylorpolynomier</strong>.<br />
<strong>Funktion</strong> <strong>af</strong> ‡ere<br />
<strong>variable</strong><br />
I Hvad er den fejl man begår ved at erstatte en funktion<br />
f med dens Taylorpolynomium P n ?<br />
I Taylors formel: For givet x …ndes et tal ξ mellem x 0 og<br />
x, så<br />
f (x) = f (x 0 ) + f 0 (x 0 ) (x x 0 ) + 1 2 f 00 (x 0 ) (x x 0 ) 2 +<br />
. . . + 1 n! f (n) (x 0 ) (x x 0 ) n<br />
+ 1<br />
(n + 1)! f (n+1) (ξ) (x x 0 ) n+1<br />
I Altså f (x) = P n (x) + 1<br />
(n+1)! f (n+1) (ξ) (x x 0 ) n+1 =<br />
P n (x) + R n (x) .<br />
Preben Alsholm<br />
<strong>Taylorpolynomier</strong><br />
De…nition <strong>af</strong><br />
Taylorpolynomium<br />
Udledning <strong>af</strong> formlen<br />
for Taylorpolynomiet<br />
Formlen for<br />
Taylorpolynomiet<br />
Eksempel 4.8.2 i<br />
Adams<br />
<strong>Funktion</strong> givet ved<br />
simpel forskrift<br />
<strong>Funktion</strong> givet ved<br />
di¤erentialligning<br />
Taylors formel med<br />
Lagrange’s restled<br />
Vurdering <strong>af</strong> fejlen<br />
ved Taylors formel I<br />
Vurdering <strong>af</strong> fejlen<br />
ved Taylors formel II<br />
Store O-notationen<br />
<strong>Funktion</strong> <strong>af</strong> ‡ere<br />
<strong>variable</strong>