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Download Abelsche und exakte Kategorien Korrespondenzen by Hans-Berndt Brinkmann, Dieter Puppe PDF

By Hans-Berndt Brinkmann, Dieter Puppe

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Ermitteln Sie, in welchen Punkten f stetig ist. 5. Untersuchen Sie, ob die Funktion f (streng) monoton wachsend oder fallend ist. a) f (x) = x4 , b) c) f (x) = x3 + 2x, √ f (x) = x − 1 f¨ ur x d) f (x) = |x2 − 2x + 1| f¨ ur x 1, 1. 6. Untersuchen Sie, ob die Funktion f in ihrem gr¨oßtm¨oglichen Definitionsbereich gerade oder ungerade ist, in den F¨allen: x3 x2 +1 , a) f (x) = b) f (x) = sin(x) · cos(x), c) f (x) = d) f (x) = 4 · sin2 (x), x2 −1 1+x2 , dabei gilt wieder sin2 (x) := (sin(x))2 . 7.

Regel von de l’Hospital: Sei I ⊂ R ein Intervall, und sei a ∈ I. F¨ ur differenzierbare Funktionen f, g : I → R gelte f (a) = 0 = g(a) (x) ur alle x ∈ I, x = a. Wenn der Grenzwert lim fg (x) und g (x) = 0 f¨ f (x) x→a g(x) existiert, so existiert auch der Grenzwert lim lim x→a x→a , und es gilt f (x) f (x) = lim . g(x) x→a g (x) Beispiel. 9 1 x − sin(x) 1 − cos(x) sin(x) cos(x) = lim = . = lim = lim x→0 x→0 6x x→0 x3 3x2 6 6 Lokale Extrema Sei I ein Intervall, und sei f : I −→ R eine Funktion. Dann hat f in x0 ∈ I ein lokales Maximum (bzw.

Eine Stammfunktion von f ist eine differenzierbare Funktion F : I → R mit F = f. 4 auch F + c eine Stammfunktion von f , wobei c eine konstante Funktion ist. • Sind F und G Stammfunktionen von f , dann ist F −G eine konstante Funktion. Denn F = f = G =⇒ (F −G) = 0 =⇒ F −G ist konstant, nach 1. 8. Beispiel. 4 1 n+1 (n x→ xn+1 . n+1 + 1)x(n+1)−1 = xn f¨ ur n ∈ N. Hauptsatz der Differential- und Integralrechnung Seien I ein Intervall, a, b ∈ I und f : I → R eine stetige Funktion. 2 beweisen kann, ist x F : I → R, x→ f (t) dt, a eine Stammfunktion von f (also F = f ).

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