Taylorpolynomier Funktion af flere variable
Taylorpolynomier Funktion af flere variable
Taylorpolynomier Funktion af flere variable
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<strong>Funktion</strong> givet ved di¤erentialligning<br />
I Find det 2. Taylorpolynomium med udviklingspunkt π 2<br />
for løsningen til di¤erentialligningen<br />
<br />
x 0 (t) = sin t + x (t) 2 π<br />
<br />
med x = 0<br />
2<br />
I Vi skal …nde<br />
P 2 (t) = x π <br />
2 + x<br />
0 π<br />
2 t<br />
π<br />
2 +<br />
1<br />
2 x 00 π <br />
2 t<br />
I Ved indsættelse <strong>af</strong> t = π 2<br />
i di¤erentialligningen fås<br />
x 0 π <br />
2 = sin π<br />
2 + x π 2<br />
2<br />
= sin π <br />
2 = 1.<br />
I Ved di¤erentiation <strong>af</strong> di¤erentialligningen fås<br />
x 00 (t) = cos t + x (t) 2 (1 + 2x (t) x 0 (t)).<br />
<br />
π 2<br />
2<br />
<strong>Taylorpolynomier</strong>.<br />
<strong>Funktion</strong> <strong>af</strong> ‡ere<br />
<strong>variable</strong><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>