Differenzieren wie Isaac Barrow

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Question statement

In seinen "Lectiones Geometricae" geht Isaac Barrow davon aus, dass man sich Kurven (vgl. Abb. unten) aus unendlich kleinen Geradenstrücken ("linelets'') zusammengesetzt vorstellen kann.

Sei $MN$ so ein unendlich kleines Geradenstück ( $M$ und $N$ also infinitesimal benachbarte Punkte) einer Kurve mit der Gleichung $f(x,y)=0$ , so darf man nach Barrow folgern, dass gilt:

$f(x+e,y+a)=f(x,y)=0$

und dass alle auftretenden $e^m,a^n$ mit $m,n>1$ zu Null werden.

Ermitteln Sie mit diesem Ansatz die Tangentensteigung $\frac{a}{e}$ für die folgende Kreisgleichung, d. h. für die implizite Funktion $f(x,y)=x^2+y^2-\var{c}^2=0$

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Es folgt zunächst:

Term1 $=x^2+y^2-\var{c}^2=0$

Durch Klammerauflösen und Unterdrücken aller übereinstimmenden Terme erhalten wir daraus:

Term2 $+a^2=0$

Durch Unterdrücken aller $e^m,a^n$ mit $m,n>1$ bleibt schließlich:

Term3 $=0$ , was uns zur Ableitung $\frac{a}{e}=$ Term4 führt.

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Andreas Vohns 4 years, 3 months ago

Andreas Vohns 4 years, 3 months ago

Andreas Vohns 4 years, 3 months ago

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Differenzieren wie Isaac Barrow

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Andreas Vohns 4 years, 3 months ago

Andreas Vohns 4 years, 3 months ago

Andreas Vohns 4 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`