Aufgabensammlung zur Vorklausur (Stand: )

Transkript

1 Aufgabensammlung zur Vorklausur (Stand: 7..08) Aufgabe : Vereinfachen Sie folgende Brüche auf einen ganzzahligen, teilerfremden Bruch oder eine endliche Dezimalzahl. Probe 0,9 0, = 0, 0, =, 0,0 =,, = : 0,7 = 8 0, : 0, = 7 0, 0, = 0, = 0,7 0,8 0 =, 0 0, 0 = Probe 0, 0,7 = 0,7 0, =,0 0,00 =, 0, 0, = :, = 7 0, :, = 7, 0,7, 0, = 0,7 7 =, ,9 0 =,9 0 0, 0 = WS 07/08 0, 0, = 0, 0, =, 0,0 =,8,7 = : 0, 8 = 0, : 0, = 0, 0, = 0, =, 0 0 =, 0, 0 = SS 08 0,7 0, = 0,0 0, = 7,07 0,0 =, 8, = : 0, = 7 0, :,8 = 0, 0, = 8 0,7 7 7 =, 0 7,8 0 =,8 0, 0 = WS 08/09 0,8 0, = 0, 0,8 = 0,89 0,0 = 8,,7 = : 0, 7, = 0,7 : 0, = 0, 7 0, = 7 0,8 =, 0 8 8, 0 =,9 0, 0 = SS 09 0, 0, = 0,0 0,8 = 9,999 0,0099 =, 0,, = :, 7 = 0,7 :, 0, = 0 0,8,7 0,7 = 0, =, 0 0,9 0 =, 0 8, 0 0 = WS 09/0 0,8 0, = 0,88 0, =, 0,0 = 0,8,, = :, = 9 0, :,7 0, = 0 0,7 7 0, = 7 0, = 0, 0 7, 0 = 8, 0,7 0 7 = SS 0 0, 0,7 = 0,8 0, =, 0,0 =,, 0,, = : 0,7 8 = 0, :, 0, = 7, 0, 0, 0, = 9 0,8 =, 0,8 0 =, 0 8, 0 7 =

2 WS 0/ 0, 0, = 0,9 0, =, 0,00 =,, 9 8 = : 0,7 = 0,7 :, 0,9 =, 0, 7,7, = 0,7 7 =,8 0 8, 0 = 7, 0, 0 = SS 0,7 0,8 = 0,8 0, = 8,888 0,00 = 0,9,, = : 0, = 7 0, :, 0, =, 0,, = 7, 0, = 8 0,7 0 = 8, = WS / 0, 0,8 = 0, 0, = 77,077 0,0 = 0, 0,9,7 = : 0,9 = 8, :, 0,9 =,7 0,,7, = 8 0, +, = 8, 0 7, 0 =, 0 0, 0 = SS 0,9 0,7 = 0,9 0, =,808 0,009 =, 0,8,, = : 0,7 8 = 0,8, : 0,7 = 0, = 9 0,, = 7 0 9,8 0 8 =, 0 0 = WS / 0, 0,8 0,8 0, :, 9 0, 0,, 0,00 0,8 0,9 8, : 0,0,, 8 0,7 9,9 0, 0,9 0 0, 0 0, 0 7 0, 0 SS 0,9 0, = 0,7 0,7 = 70,7 0,0 = 0,9 0,7, = : 0, 7 = 0, 0,8 : 0,9 =,,8 8, = 0,9 7 0,8 = 0, 0 0, 0 = 0,8 0, 0 7 = WS / 0, 0,7 = 0, 0, =, 0,0 = 0, 0, 0, = : 0,08 8 = 0,8,8 : 0,7 9 =,8 0, 8,9 = 7 0,8 =, 0, 0 = 0, 0 0,9 0 7 = SS 0, 0, = 0, 0,8 = 7,07 0,0 = 0,8,, = :, = 0,7 :, 0, = 7, 0, 0, 0, = 8 0, +, =, 0,8 0 =,8 0, 0 = WS / 0, 0,7 = 0,88 0,8 = 0, 0,007 = 0,8, 0,7 = : 0, 9 = 0,7 0, :,9 =,0 0,8 = 0,7 + 7 =, = 0, =

3 SS 0, 0, = 0,9 0, =, 0,0 =, 0, 8 = : 0,7 0 =,8 0, :, 7 = 0,, = 0, 0,0 = 78 0, 0 =, = WS / 0,98 0,7 = 0,7 0, = 0, 0, = 0,7 0,,, = : 0,8 = 0,8 0,8 :,7 = 7, = 7, 0,, 0,7 = 0, =, 0 0 = SS 0,78 0, = 0,8 0, =, 0, =, 0,,, = :, = 0,7 0, :,9 = 7,8, =, 0,, 0, = 0,8 0 0,8 0 =, 0 0 = WS /7 0, 0,9 = 0, 0,7 = 707,07 0, = 0,7 0,9 0,7 = : 0,8 =,,7 : 0,7, = 9, = = 0, 0 7, 0 =, = SS 7 0, 0,8 = 0, 0,7 = 0,0 0,8 = 0,,,7 = : 0,0, =, 0 : 0, =,, 9 =,8,,0 0, = 0,7 0 9,9 0 7 = 0, = WS 7/8 0,8 0, = 0, 0, =, 0,0 = 0,8,, = : 0,07 8 = 8, :, 0,9 =,,8 8, = 8 0, +, = 9 0, 0 = 0, 0 0, 0 = SS 8,0 0,8 = 0, 0,7 =,00 0,0 = 0,, 8 0,7 = : 0, 0,9 = 0,,7 : 0,, = 7, 9, 8 =, 0,7,, = 7, =,9 0 0 = WS 8/9 0,9 0,7 = 0, 0,8 =,0 0, =, 0,8,, = : 0,08 8 = 0, 0,7 :, 8 = 0, 0, 0, 0, = 0,8 0,0 +, 0, = 0, 0 0, 0 =, = Aufgabe : Bestimmen Sie sofern eistent die Nullstellen folgender Funktionen. Probe f () = f () = + 8 f () = ( 7 ) f () = ( ) ( ) ( ) ( ) f () = 7 ( + ) + +

4 Probe f () = + 0 f () = + f () = ( ) f () = ( ) ( ) ( ) ( ) f () = + ( + ) + WS 07/08 f () = f () = f () = e ( ) f () = ( ) ( ) ( ) ( ) f () = 7 ( + ) + + SS 08 f () = + f () = f () = ( ) f () = ( ) ( ) ( + ) ( ) f () = + ( + ) + + WS 08/09 f () = + 0 f () = + f () = ( ) f () = ( ) ( ) ( ) ( ) f () = + ( + ) + SS 09 f () = 7 + f () = 7 + f () = e + ( + ) f () = ( 8) ( ) ( ) ( ) f () = + ( + ) + + WS 09/0 f () = f () = + f () = ( + 8) f () = ( + ) ( + ) ( + ) ( + ) f () = + ( ) + SS 0 f () = 9 + f () = 0 f () = ( ) f () = ( ) ( ) + ( ) ( ) f () = + ( ) + WS 0/ f () = + + f () = f () = e + ( ) f () = ( ) + ( + 9) ( ) ( ) f () = + + ( ) SS f () = + 0 f () = + f () = + ( ) f () = ( + ) ( ) ( + ) ( + ) f () = + 0 ( ) WS / f () = + f () = f () = sin( ) ( + ) f () = ( + ) ( + ) + f () = ( ) SS f () = f () = + + f () = ( ) f () = ( + ) ( ) ( + ) ( ) + f () = ( ) + WS / f () = + + f () = + + f () = e ( + ) f () = ( + ) ( ) ( + ) ( + ) f () = + + SS f () = + f () = + f () = ( + ) f () = ( ) ( + ) ( ) ( ) f () = + + WS / f () = f () = + f () = ( + ) f () = ( ) ( ) + ( ) ( ) ( ) f () = + 9 ( + ) + SS f () = + f () = + f () = e sin ( ) ( ) f () = ( ) ( + ) + ( ) ( ) ( ) f () = + +

5 WS / f () = f () = + 8 f () = cos ( ) ( + ) f () = ( + ) ( + ) ( + + ) ( + ) ( + ) f () = SS f () = + f () = f () = ( + ) f () = ( ) ( ) + ( + ) ( ) ( ) f () = ( ) ( ) WS / f () = + + f () = + + f () = e ( + ) f () = ( + ) ( ) + ( + ) ( + ) ( + ) f () = ( ) + SS f () = + + f () = + f () = ( ) f () = 9 ( + ) ( ) ( ) ( + ) ( ) f () = ( + ) ( ) WS /7 f () = + f () = 8 f () = ( ) f () = 9 ( ) ( + ) ( ) f () = + + SS 7 f () = 7 + f () = + 8 f () = ln() ( 8) f () = ( + ) ( ) ( + 8) ( ) ( ) f () = WS 7/8 f () = f () = + + f () = ( ) f () = ( ) ( + ) ( ) ( ) ( ) f () = SS 8 f () = + 0 f () = + + f () = ln( ) ( + ) f () = ( ) ( ) ( + ) ( ) ( ) f () = + + WS 8/9 f () = + + f () = + f () = ( + ) f () = ( ) ( + ) ( ) ( ) ( ) f () = ( ) Aufgabe : Lösen Sie die folgenden Gleichungen nach auf (alle in den Rechnungen auftauchenden Ausdrücke seien definiert). Vereinfachen Sie die Lösung soweit wie möglich. Probe e = a + = a a ( 7 + ) + b 7 = c a ( + ) + b = c a + b = c Probe e = a = a a + b ( ) = c a ( + ) + b ( ) = c a + + b = c WS 07/08 e = a = a a + b ( ) = c a + b ( ) = c a + b + = c SS 08 e = a = a a ( ) + b ( + ) = c a ( + ) + b ( ) = c a + + b = c WS 08/09 e = a = a a ( ) + b ( ) = c a ( ) + b ( ) = c + + a = b SS 09 ln() = a 0 = a a ( ) + b ( ) = c a ( ) + b ( ) = c e a + e + a = e b

6 WS 09/0 e = a = a a ( ) + b ( ) = c a ( ) + b ( + ) = c + = a SS 0 e = a = a a ( 8 ) b ( 8 ) = c a + b = c + a + + b = c WS 0/ e a+ b = c = a a ( a) + b ( b) = ab a + b + = a + b + a + b = c SS e = a a b = c a (7 + b) + a (9 b) = a + = a e a = e + WS / e + = a = a a ( + b) b ( + a) = a b + + = = + SS e + = a a = b a ( + b) b ( + a) = a b + = 8 = + a WS / e a = b = a ( b) + b ( a) = a + b + + = = SS e a = = 8 a ( b) + b ( a) = a + b 0 + = 8 = + WS / e = + = 9 a ( + b) b ( + a) = a b + + = 9 = + SS a e b = = a a ( b) + b ( a) = a + ab + b + + = a 7 = + WS / a e + = b a = e b a ( + b) b ( + a) = 9a ab + b + = 8 = SS a e b = ( ) a = b a ( b) + b ( + a) = a + ab b = + = 8 + WS / e a = b = a ( b ) b ( ab) = 9a b + = ( + ) ( +) ( ) = ( + ) SS a e b = 8 = a ( + b) b ( a) = a + b + = + + = WS /7 e a b = + = 7 a ( b) + b ( + a) = 8a + 0ab + b = 0 e + e = e SS 7 WS 7/8 SS 8 + a + e = b = a ( b) b ( a) = a + ab b + 8 = + = 9 + e + a + b = + = a ( b) + b ( a) = a + ab + b + = e e = e + b + a = + ( ) = + 7 = ( 9 )( +) a ( + b) b ( a) = 9a ab + b + =, Tipp: y = WS 8/9 e = a e = e a ( b) + b ( a) = a + ab + b + + = = 7 +

7 Aufgabe : Bestimmen Sie die Lösungsmengen folgender Gleichungen. Probe ( + ) = ( + ) ( + ) / = 8 / = 9 + = = Probe ( + ) = ( + ) ( ) / = 9 8 / = = WS 07/08 ( ) = ( ) ( ) / = 8 / = = + = = SS 08 ( ) = ( ) ( + ) / = / = = 7 = WS 08/09 ( ) = ( ) ( + ) / = 8 8 / = + = = SS 09 ( + ) 0 = ( + 7) 0 ( ) / = 8 /7 = = 7 = WS 09/0 ( 7) 8 = ( + ) 8 ( + 8) / = / = 8 7 = = SS 0 ( ) = ( + 7) ( ) = / = 0 + = 9 8 = WS 0/ ( + ) = ( + ) ( + 9) / = 8 / = + = = 8 SS ( ) = ( ) ( ) / = 80 / = = = WS / ( + ) 0 = ( + ) 0 ( + ) / = 8 / = + = =

8 SS ( ) = ( ) (8 ) / = / = 8 + = = WS / ( + ) = ( 7) ( + 8) / = 9 / = 7 + = = 8 SS ( ) 0 = ( ) 0 ( + ) / = 8 / = = = ( )/ WS / ( + 7) 8 = ( + ) 8 ( ) / = 8 / = = = SS ( ) = ( 7) (8 + 9) / = 9 / = = = WS / ( + ) 0 = (8 + ) 0 ( 8) / = 7 / = + 8 = = SS (0 ) = (7 ) ( + ) / = / = = = WS / ( + ) = ( + ) ( + ) / = / = = = SS ( + ) 0 = ( ) 0 ( + ) / = / = = = WS /7 ( 7 + 7) = ( + 8) ( + 8) / + = 9 / = 0 + = = SS 7 ( + 7) 0 = ( ) 0 8 ( 0) / = 8 = 8 + = =

9 WS 7/8 ( ) 0 = ( ) 0 ( + ) / + = 9 / = = = SS 8 ( + 7) 8 = ( + ) 8 ( ) / + = 9 8 / = + 0 = = 8 0 WS 8/9 ( 7) = ( ) ( ) / + = 0 / = 8 + = = Aufgabe : Bestimmen Sie jeweils die erste Ableitung der folgenden Funktionen. Probe f () = e + ( ) f () = 8 + f () = sin( ln(e + + ) ) f () = Probe f () = e sin ( ) ( ) f () = + f () = cos(e ln() ) f () = WS 07/08 f () = e + ( ) f () = + f () = sin( ln(e + + ) ) f () = ( + ) SS 08 f () = e ( ) f () = + f () = cos(e ln( ) ) f () = ( + e ) WS 08/09 f () = e ( + ) f () = + f () = sin(e +sin( ) ) f () = + ( + ) SS 09 f () = e ( ) f () = + f () = cos( ) f () = + WS 09/0 f () = e ( ) ( + ) f () = + f () = sin(e sin (e ) ) f () = SS 0 f () = e g () g() f () = + f () = ln( + ) ( + ) ( ) f () = ln( + ) f () = ln( +) ( ) f () = ln(e +) (+) f () = ln( ) f () = ln( + ) ( + ) f () = ln( + ) ( + ) f () = ln(g()) (g()) f () = sin(sin(e + )) f () = +

10 WS 0/ f () = e ( + ) f () = ( + ) f () = sin(ln( + )) f () = SS f () = e + ( + ) f () = sin( ) f () = g () g() f () = e e + WS / f () = e ( ) f () = g() f () = ln(e + ) ( + ) f () = ln(e + ) ( + ) f () = ln( ) f () = e g ( ) g() f () = g ( ) SS f () = e sin ( ) sin(e ) f () = ( ) f () = cos( ln( sin ( ) ) ) f () = WS / f () = e + ( + ) n f () = ( ) n f () = ln( + ) f () = ln( + ) ( + ) f () = sin( e g ( ) ) f () = SS f () = e a ( + ) f () = n ( + ) f () = sin( ln( e + ) ) f () = WS / f () = e a ( ) f () = ( + ) 7 f () = ln( e ( + ) ) f () = ln( a ) f () = sin( ) f () = ( +) ( +) SS f () = e a (a ) f () = n + f () = ( sin( ln( +) + ) ) f () = WS / f () = e a+ b (a + b) f () = f () = ln( e + ) f () = ln( e ) f () = cos( ln( g () ) ) f () = + SS f () = e a ( a ) f () = n ( + ) f () = ln( e + + e e + e ) (vereinfachen) f () = ( sin( g () ) ) WS / f () = e g () g() f () = f () = +e f () = sin( ln( e + ) ) f () = sin ( ) f () = ln( a ( + ) e ) (vereinfachen) SS f () = sin() e f () = ( + ) n f () = ln( a e ) (vereinfachen) f () = sin( ln ( +) ) f () = (sin()) sin( )

11 WS /7 f () = e + f () = n ( + ) f () = ln( a e + ) (vereinfachen) f () = (e cos ( + ) ) f () = g() ( a+b) SS 7 f () = e a f () = f () = ln( e + ) (vereinfachen) ( + ) n a + f () = sin(e sin( g ( ) ) ) f () = (a + b) g ( ) ln( + ) ( + ) WS 7/8 f () = + f () = f () = ln( ) (vereinfachen) ( + ) e a + f () = sin( ln ( )) f () = ln ( ) SS 8 f () = e + (+) + f () = f () = ln( a ) (vereinfachen) ( ) n b a + f () = sin( (e + + ln( + )) ) f () = ( f '() = 0? ) WS 8/9 f () = e a +b (a + b) f () = n ( ) e + f () = ln( ) (vereinfachen) a + f () = sin( ln( sin( + ) ) ) f () = g() g ( ) Aufgabe : Bestimmen Sie sofern eistent die globalen Etrema der folgenden Funktionen. f l () = Probe f () = Probe f () = WS 07/08 f () = SS 08 f () = WS 08/09 f () = für 0 f r () = 9+ für > 0 f l () = + für 0 < f r () = + + für f l () = + 9+ für 0 f r () = +7 + für > f l () = e für 0 < f r () = + f () = + für f () = 9 + für < <. f () = + f l () = + SS 09 f () = f l () = WS 09/0 f () = + SS 0 f () = für f r () = e + für > für 0 f r () = + + für > 0 f () = e für 0 f () = für >

12 f () = WS 0/ f () = f () = SS f () = WS / SS 09 SS f () = f () = WS / f () = SS f () = für 0 f () = + für > 0 + für < f () = ( ) e für f () = ( ) e + für 0 0 f () = + + für > + für < 0 f () = e für 0 f () = + f () = + für < < f () = + + f () = WS / f () = + für < f () = ( ) e für SS f () = f () = WS / f () = 0 f () = ++ für 0 < < f () = e f () = ( ) e + für 0 f () = SS f () = ( ) f () = WS / f () = + SS f () = 0 + für >, Hinweis: e 0 0 f () = ( + ) ( ) für 0 < < f () = ( ) e ( ) für 0 f () = ( ) e für > f () = f () = + WS /7 f () = SS 7 f () = 0 f () = ( ) für 0 < < f () = ( ) e ( ) für 0 < f () = (8 ) e für f () = WS 7/8 f () = f () = ln( + ) 0 < e f () = (e ) e e für e

13 f () = + 9 SS 8 f () = 0 + f () = + für 0 < f () = ( ) e > WS 8/9 WS / Aufgabe 7: Bestimmen Sie die Lösungsmengen der folgenden in Tableauform gegebenen linearen Gleichungssysteme. Probe y z r.s. 0 7 Probe y z r.s. 0 0 WS 07/08 y z r.s SS 08 y z r.s. WS 08/09 y z r.s. 7 7 SS 09 y z r.s. 8 WS 09/0 y z r.s. 0 8 SS 0 y z r.s. WS 0/ y z r.s. SS y z r.s. 7 WS / y z r.s. 9 0 y z r.s. 0 0 y z r.s. 0 8 y z r.s. 0 0 y z r.s. 9 7 y z r.s. 0 y z r.s. 8 y z r.s y z r.s. y z r.s. 8 9 y z r.s. 7 y z r.s. 0 y z r.s y z r.s. 0 0 y z r.s y z r.s. y z r.s. 0 8 y z r.s. 0 y z r.s y z r.s. 0 y z r.s y z r.s. 0 7 y z r.s. 0 0

14 SS y z r.s WS / y z r.s. 9 SS y z r.s. 7 0 WS / y z r.s. 0 SS y z r.s. 7 7 WS / y z r.s SS y z r.s WS / y z r.s. 7 0 SS y z r.s. 7 0 WS /7 y z r.s. SS 7 y z r.s. 0 7 WS 7/8 y z r.s. 0 0 SS 8 y z r.s. y z r.s y z r.s. 8 0 y z r.s. y z r.s. 0 9 y z r.s. 7 y z r.s. 8 y z r.s y z r.s y z r.s y z r.s. 0 y z r.s. 9 y z r.s. y z r.s. 7 y z r.s y z r.s y z r.s. y z r.s. 8 y z r.s. 7 7 y z r.s. 0 0 y z r.s. y z r.s y z r.s. 7 y z r.s. 7 8 y z r.s. 7 y z r.s. 9 y z r.s. 8

15 WS 8/9 y z r.s. 0 7 y z r.s. 0 0 y z r.s. 8 Aufgabe 8: Bestimmen Sie die Lösungsmengen der beiden folgenden, nichtlinearen Gleichungssysteme. Probe y = +y +, + = = +, y = + y Probe y = 9 y, + y = + =, + y = WS 07/08 + y y = 7, 9 = + =, + y = 8 y SS 08 + y =, ( ) + y = WS 08/09 + y =, SS 09 + y =, WS 09/0 + = y, SS = y, WS 0/ + y = 7, SS + y =, WS / = y +, SS + y = 8, WS / + y =, SS y = 7, WS / + y =, SS + y = 0, WS / y =, SS y = 7 +, WS / + y = 0, SS y = +, WS /7 = 7 + y, y + y =, + = 9y y + + = =, + y = y y y + + = y y + + y = = = = y + =, + y = 8 y + + =, = y + y + + =, y = + =, y + = y + + =, +y = 7 y + y + y + = y =, y + + y = y + y + 9 = + 8 =, + y = 0 y y y + 8 = ( ) + = 7, y = + y + y + y + = y y + = 7 + =, (y + ) = + y y + + =, y + = y + y y 8 = + y + =, y y + = y + y + = y 8 y = + =, y + = y + y y + + = 7, y = y + y + + y + 7 = y + =, y = y y + 7 SS 7 y + = 9, = 8 y = y + =, y y + = y y = y + + = y, y + = y 8 y y =, + y = 0

16 WS 7/8 + y =, SS 8 y + = WS 8/9 y + = y y + = y + =, + y + + y + = y + 8 y = y + y + = 7 y + + =, + y + + y = 0 y + + = 7, + y + y = Lösungen Bei allen Aufgaben muss der Rechenweg erkennbar sein (auch beim Bruchrechnen mindestens Zwischenschritt). Ohne Rechnung gibt es auch bei richtigem Ergebnis keine Punkte. Aufgabe Probe,, 0 0, 0, 0 Probe 0,8, WS 07/08,, SS 08,, WS 08/09 0,8, SS 09,, WS 09/0,7, SS 0 0,9, WS 0/ 0,9, SS 0,9, WS / 0,7, SS,, WS / 0,7, SS,, WS / 0,, , 0 SS 0,9, WS / 0,9, SS,, WS /,, SS,, WS /7 0,, 00, 0, 0 70 SS 7 0,7 0, 0,0 0 0 WS 7/8 0,7,7 70, SS 8,, 0,0,, WS 8/9, 00,

17 Aufgabe Probe {, } {, 0, } {, } Probe {, } {0,, } ± {, } WS 07/08 {, } {, 0, } ± {, } SS 08 {, } {, 0, } ± WS 08/09 {, } {, 0, } ± 8 SS 09 {, } {, 0, } ± {0, } WS 09/0 {, } {, 0, } ± SS 0 {, } {, 0, } {0, } WS 0/ {, } {, 0, } {, 0, } SS {, } {, 0, } {, 0, } WS / {, } {, 0, } {, } SS {, } {, 0, } {0, } WS / {, } {, 0, } {, 0} 0 SS {, } {0,, } ± 0 WS / {, } {0,, } {, 0} 0 SS {, } {, 0} ± 0 WS / {, } ± 0 SS {, } {, 0, } 0 WS / {, } 0 {, 0} {0, } SS {, } {, 0, } ± 0 WS /7 {, } {, 0, } 0 SS 7 {, } ± {, } {0, } WS 7/8 { ; } { ; 0; } ± 0 0 SS 8 {, } {, 0, } ± 0 WS 8/9 { ; } {0; ±} ± Aufgabe Probe Probe ± ln(a) + WS 07/08 SS 08 WS 08/09 ( ln(a) + ) ln(a) ln() 7 c a a + b ln(a) + ln(a) ln() ln(a) + ln(a) ln() ± ln(a) ln(a) ln() + c + b a + b b + c a + b c + a b a + b ( + ln(a) ln() ) c + a + b a + b SS 09 e a + ( + ln(a) ln(0) ) c + a + b a + b WS 09/0 + ln(a) ( + ln(a) ln() ) ± c + a + b a + b SS 0 ± + ln( + a) + ln(a) ln() ± 8 a b + c a b ln( c a a + b ) ln() ln( c a + b ) a + b ln() ln( b + c a + b ) ln() ln( c a + b ) a + b ln() ln( c + a + b a + b ) ln() ln( c + a + b a + b ) ln() c + a + b ln( ) a + b ln() c ln( a b/ ) ln() c ln( a/ + b ) } ln() c ln( a + b ) ln() c ln( a + b ) ln() c ln( 9a + b ) ln() b ln( + a ) ln() b ln(e a + e a ) ln( a ) ln() c ln( a + b ) ln()

18 WS 0/ a ( ln(c) b ) ± ln(a) ln() SS ± ln(a + ) + a ( b + ln(c) ln() ) ± a WS / ln(a ) ± + ln(a) ln() a + b ( c a b ) a ln() ln() a + b a SS + ln(a) log a ( + b) WS / ± ln(b + ) a ± a + b a a + b SS + ln( + a ) ± a + b WS / ± ln() ± ± a + b SS a ln( + b ) ± log (a) ± a + b a log () WS / ln(a(b + )) ln(a) b a b SS b ln(a ) b /a = a b a b WS / a + ln(b + ) ± a + b {, 0, } SS ± b ln(a ) ± a b WS /7 ± a + ln(b + ) ± a + b ± SS 7 + ln( a + b + ) a + b WS 7/8 ln((a + )(b + )) ± a + b 0 SS 8 b ln( ) a {, 0} ± a b ± WS 8/9 ln(a+) ± ln() + a + b ± Aufgabe Probe {, } ± 8 Probe {, } {, } WS 07/08 ± ± 8 0 SS 08 ± ± WS 08/09 ± ± 0 ± 8 SS 09 {, } {, 0} ± 8 7 WS 09/0 {, } { 0, } 8 SS 0 {, } {, } ± 8 7 ± 8 WS 0/ {, } {, } ± 8 ± 8 SS {, } {, } ± WS / {0, } {, } ± SS {, } {, } 9 WS / {, } {, 0} ± 8 SS ± {, } ± 9 ± WS / {, } {, } ± 8 SS {, } {, } 0 WS / {, } {, 0} ± 7 9 ± 8 SS {, } {, } ± ± 8 WS / ± {, } SS {, } {, } ± ± 8

19 WS /7 {, } { 0, } 9 ± SS 7 {, } {, 7} WS 7/8 {, } {, 0} { ; 0} ± SS 8 {, } {, 0} ± 8 ± WS 8/9 {, } {, } 9 ± 8 Aufgabe Probe f'() = f () + e + ( ) f'() = (8 + ) / + ( + ) ln( + ) ( + ) f'() = ( + ) f'() = cos( ln(e + e + ) ) + e + + f'() = + f () ln() Probe f'() = f () cos() + e sin ( ) ( ) f'() = ( + ) / ( ) ln( + ) ( ) + f'() = ( ln( + ) ) f'() = sin(e ln() ) (e ) f'() = ln() WS 07/08 f'() = f () + e + ( ) f'() = ( + ) / + ( ) ln( + ) ( ) f'() = ( ) 0 f'() = cos( ln(e + + ) ) e + e + + f'() = f () ln() + ( + ln()) SS 08 f'() = f () + e ( ) f'() = ( + ) 7/ f'() = e e + ( + ) ln(e + ) ( + ) ( + ) f'() = sin( e ln( ) ) e ln( ) ( ln() + ) f'() = f () ln() + ( + e ) WS 08/09 f'() = f () ( ) + e ( + ) f'() = ( + ) / f'() = ln() oder f'() = ln() f'() = cos(e +sin( ) ) e +sin( ) ( + cos() ) f'() = f () ln() + + ( + ) SS 09 f'() = f () ( ) + e ( ) ( ) f'() = ( + ) / f'() = + ( + ) + ln( + ) ( ) ( + ) + ( + ) ln( + ) ( + ) f'() = ( + ) oder f'() = sin( ) ( + ln() + ) f'() = f () ln() 0

20 WS 09/0 f'() = e ( ) ( ) ( + ) + e ( ) ( + ) f'() = ( + ) / f'() = ln() + ( + ) + ln( + ) ( ) ( + ) oder f'() = ln() + ( + ) ln( + ) ( + ) ( + ) f'() = cos(e sin (e ) ) e sin (e ) cos(e ) e f'() = ln() ln() SS 0 f'() = e g () g'() g() + e g () g'() f'() = ( + ) / f'() = g'() g() (g()) + ln(g()) ( ) (g()) g'() f'() = g'() g() (g()) ln(g()) g() g'() (g()) f'() = cos(sin(e + )) cos(e + ) (e + ) oder f'() = f () ln() ( + ln()) WS 0/ f'() = f () ( ) + e ( + ) ( ) f'() = ( + ) 8/ f'() = e e + ( + ) + ln(e + ) ( ) ( + ) oder e e + ( + ) ln(e + ) ( + ) f'() = ( + ) f'() = cos(ln( + )) ln() + f'() = f () ln() ln() SS f'() = f () + e + ( + ) f'() = (sin( )) / cos( ) f'() = e + e + ( + ) + ln(e + ) ( ) ( + ) oder f'() = e + e + ( + ) ln(e + ) ( + ) ( + ) f'() = f () ln() g'() + g () g() g'() f'() = f () e + WS / f'() = f () ( ) + e ( ) ( ) f'() = g() 8/ g'() f'() = + ln() f'() = f () g'() + e g ( ) g() g'() f'() = f () ln() g ( ) ln() g'() SS f'() = f () cos() + e sin ( ) cos(e ) e f'() = ( ) / f'() = + f'() = sin(sin( ) ln()) cos( ) ln() f'() = f () ( ln() + ) WS / f'() = f () + e + n ( + ) n f'() = n ( ) n/ + ( + ) ln( + ) ( + ) f'() = ( + ) f'() = cos( e g ( ) ) e g ( ) g() g'()

21 f'() = f () ( ln() + ) SS f'() = f () a + e a ( + ) f'() = n ( + ) /n f'() = + + f'() = cos( ln( e + ) ) e e + f'() = f () ( ln() + ) WS / f'() = f () a + e a ( ) f'() = 7 ( + ) / f'() = ln() f'() = f () ln() cos( ) ln() f'() = f () [ ln( +) + ( +) + ] SS f'() = f () ( ) + e a (a ) ( ) f'() = ( n + ) / n n f'() = f'() = ( sin( ln( +) + ) ) cos( ln( +) + ) + f'() = f () [ ln() ln() + ] WS / f'() = f () a + e a+ b (a + b) a f'() = f () ln() ( ) f'() = ln() + f'() = f () ln() (+) () f'() = sin( ln( g () ) ) g' g () SS f'() = f () + e a f'() = n ( + ) /n f'() = 0 f'() = ( sin( g () ) ) cos( g () ) g () ln() g'() f'() = f () [( + e ) ln() + ( + e ) ] WS / f'() = f () g' () + e g () g' () f'() = (/ + ) / / f'() = ln() f'() = cos( ln( e + ) ) e e + f'() = f () [cos() ln() + sin() ] SS f'() = [sin() e + cos() e + sin() e ] f'() = n ( + ) n/ f'() = f () ln() cos( ln( + ) ) + f'() = ln() f'() = f () [cos() ln(sin()) + sin() cos() sin() ] WS /7 f'() = (e + e ) ( + ) e ( + ) f'() = ( n ( + ) /n ) ( + ) f'() = + f'() = f () [a ln(g()) + (a + b) g'() g() ] f'() = f () ( sin( + )) ( + )

22 SS 7 f'() = f () a + e a ( /) / f'() = ( n/) ( + ) n/ ( + ) f'() = + + f'() = cos(e sin ( g ( ) ) ) e sin( g ( ) ) a cos( g () ) g'() f'() = f () [g'() ln(a + b) + g () a + b ] ( + ) ln( + ) WS 7/8 f'() = ( + ) f'() = ( ) ( + ) / ( + ) f'() = + ln() + f'() = f () ln() cos( ln()) (ln() + ) f'() = f () ln() SS 8 f'() = ( f () e + ( + ) ) oder f'() = f () ( ) f'() = ( n ) ( ) n/ ( ) f'() = b + a ln() + f'() = cos( (e + + ln( + )) ) (e + + ln( + )) (e ) f'() = f () / [ ln() + ] = 0 = e WS 8/9 f'() = f () a + e a +b a f'() = ( n ) ( ) /n ( ) f'() = + f'() = f () [ g'() ln( g()) + g() g'() g() ] f'() = cos( ln( sin( + ) ) ) cos( + ) sin( + ) Aufgabe "Kandidaten" globales Maimum globales Minimum Probe (, ), (, ), (0, ), (, ) (, ) Probe (0, ), (, ), (, ), (, ) (0, ), (, ) (, ) WS 07/08 (0, ), (, ), (, ), (, ) (, ) (, ) SS 08 (0, 0), (, ), (, ), (, ), (, e) (, ) (0, 0) WS 08/09 (, ), (, ), (, ), (, ), (0, ) (0, ) (, ), (, ) SS 09 (, ), (, ), (, ), (, ), (, e) (, e) WS 09/0 (, ), (, 0), (0, ), (, 9) (, 9) SS 0 (0, 0), (, ), (, ), (, ), (, e), (,,) (0, 0) WS 0/ (, ), (, 0), (0, ), (0, ), (, ) (0, ), (, ) SS (, ), (, 0), (, ), (, 0), (, ), (, ) (, ) (, ) WS / SS 09 SS (0, ), (, ), (, ), (, 0), (,,) (,,) (, 0) WS / (, 0), (, 0), (0, 0), (, ), (, ) (, ) (, ) SS (, ), (, 0), (, ), (, ), (, ), (0, ) (0, ) WS / (, ), (, 0), (, ), (, 0), (, ) (, ) (, 0) SS (, ), (, ), (0, ), (, ), (, + ) WS / (0, ), (, ), (, ), (, 0), (, e ), (, e ) (, e ), (, e ) (, 0) SS (, ), (, 0), (, 0), (, ) (, ) WS / (, ), (, 0), (0, ), (, ), (, ), (, 0) (, ) SS (, ), (0, ), (, 0), (, ), (, 0), (, ) (, )

23 WS /7 (, ), (, 0), (0, ), (, 0), (, ) (0, ) SS 7 (, ), (0, ), (, 0), (, ), (, 0), (, ) (, 0) WS 7/8 (0, ), (, 0), (e, 0), (e, ) (0, ), (e, ) (e, 0) SS 8 (, ), (, 0), (0, 9), (0, ), (, ), (, ), (, ) (0, ) (, ) WS 8/9 WS / Probe Probe WiSe 07/08 SoSe 08 WiSe 08/09 e SoSe 09 WiSe 09/0 SoSe 0 WiSe 0/ SoSe 9 e e SoSe WiSe / SoSe WiSe / SoSe WiSe / SoSe WiSe / SoSe WiSe / SoSe 7 WiSe 7/8 SoSe 8 9 Aufgabe 7 Probe viele (0,, ) Probe viele (,, ) WS 07/08 (,, ) viele SS 08 viele (0,, ) WS 08/09 viele (,, ) SS 09 viele (,, ) WS 09/0 (,, 0) viele SS 0 (, 0, ) viele WS 0/ (,, ) viele SS (,, 0) viele

24 WS / viele (, 0, ) SS viele (,, 0) WS / (,, ) viele SS viele (,, ) WS / (,, ) viele SS viele (,, ) WS / viele (,, ) SS (0,, ) (, ) WS / (, 0, ) viele SS viele (,, ) WS /7 viele (,, ) SS 7 viele (,, ) WS 7/8 (,, ) viele SS 8 viele (,, ) WS 8/9 (0,, ) viele Aufgabe 8 Probe {(0, ), (, )} {(, ), (, )} Probe {(0, ), (, )} {(, ), (, )} WS 07/08 {(, ), (, )} {(, ), (, )} SS 08 {(, ), (, )} {(, ), (, )} WS 08/09 {(0, ), (, )} {(0, ), (, )} SS 09 {(, ), (, )} {(, ), (, )} WS 09/0 {(, ), (, )} {(, ), (, )} SS 0 {(, ), (, )} {(, ), (, )} WS 0/ {(, ), (, )} {(, ), (, )} SS {(, ), (, )} {(, ), (, )} WS / {(, )} {(, )} SS {(, ), (, )} {(, )} WS / {(, )} {(, ), (, )} SS {(, ), (, )} {(, ), (, )} WS / {(, )} {(, )} SS {(, )} {(, )} WS / {(, )} {(, )} SS {(, )} {(, )} WS / {(, )} {(, )} SS {(, )} {(, )} WS /7 {(, )} {(, )} SS 7 {(, ), (, )} {(, )} WS 7/8 {(, )} {(, )} SS 8 {(, )} {(, )} WS 8/9 {(, )} {(, )}