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)UHLH XQG +DQVHVWDGW +DPEXUJ :LUWVFKDIWVJ\PQDVLHQ %HK|UGH IU %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ $ELWXU /HKUHUPDWHULDOLHQ ]XP /HLVWXQJVNXUV 0DWKHPDWLN 0D/./0: 6HLWH YRQ $1$/<6,6 , )XQNWLRQHQVFKDU H[SRQHQWLHOOHU )XQNWLRQHQ *HJHEHQ LVW GLH 6FKDU YRQ )XQNWLRQHQ I D GXUFK [ D I [ [ D H [ D = + D 8QWHUVXFKHQ 6LH GLH *UDSKHQ YRQ I D DXI 6FKQLWWSXQNWH PLW GHQ .RRUGLQDWHQDFKVHQ $V\PSWRWHQ ([WUHPSXQNWH :HQGHSXQNWH 6\PPHWULH E =HLFKQHQ 6LH GHQ *UDSKHQ YRQ I IU ± [ F %HVWLPPHQ 6LH HLQH )RUPHO IU GLH QWH $EOHLWXQJ YRQ I D (UPLWWHOQ 6LH HLQH 6WDPPIXQNWLRQ ) D ]XU I D 6LH N|QQHQ GD]X GDV HEHQ HUKDOWHQH 5HVXOWDW YHUZHQ GHQ ZHQQ 6LH GLHV WXQ EHJUQGHQ 6LH ,KU 9RUJHKHQ G (UPLWWHOQ 6LH GLH *OHLFKXQJ GHU 2UWVNXUYH DOOHU ([WUHPSXQNWH GHU *UDSKHQ YRQ I D H 'HU *UDSK YRQ I XQG GLH [$FKVH EHJUHQ]HQ HLQH QDFK UHFKWV LQV 8QHQGOLFKH UHLFKHQGH )OlFKH %HUHFKQHQ 6LH GHQ ,QKDOW GLHVHU )OlFKH

PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

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Page 1: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−= + ⋅ ∈ ∈

≤ ≤

Page 2: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= =

−= = −

→ ∞ + −

→ ∞ →

→ −∞

′ = − −

′′ = + −

′ =− −′′ − = − < ∈

−−

′′ =

→ +∞ ′′′ ≠′′ −

−−

− =

− + ⋅ ≠ + ⋅

− = −

− + ⋅ ≠ − + ⋅

Page 3: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

( ) ( )( )

< > −

= + −

′ = − + −

′′ = + −′′′ = − + −

= + −

−− −

′ = = + ∈( )< > −= − ⋅ + −

−= + ⋅

Page 4: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−= − + +

′ =

−−

= − = −

−=

−=

−= ⋅

( )− − −

→∞ →∞ →∞⋅ = − ⋅ + = − + + =

−= ⋅

Page 5: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= − + + ≥

= +−

= ⋅

Page 6: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−−⋅=−

Page 7: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= + + +

′ = + +′′ = +

= == + + + == + + + =

′′ = + =

⇔ − ⇔ −− −

= − + +

=

= ⋅ =

= − = − + − −

− + − − =

− + + =

Page 8: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−− −

− −

( )− + + = − ⋅ − −

− − =

= + ≈

= − ≈ − ∉

′ ′′= ∧ <′ = − + −′′ = − +

− + − =

− + =

= ± −

≈≈

′′ = − <′′ = >

=

Page 9: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= +

−= ⋅ = ⋅

→∞⋅ =

===⋅→∞→∞

→∞

Page 10: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= ∧ <

= ⋅ − −

′ = − ⋅ −

′′ = − ⋅

′′ = − <≈

Page 11: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= − ∈ >⋅

• ∞

∈ �

Page 12: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

{ }=

→ +∞ → −∞

− = ⇔ = ⋅ ⇔ =⋅

− ⋅′ = +⋅

⋅′′ = −⋅

− ⋅′′′ = +⋅

′ = ≠

− ⋅ + = ⇔ = ⇔ =⋅

′′ =⋅

′′ >

Page 13: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

′′ = ≠

⋅ − = ⇔ = ⇔ =⋅

→ +∞ −′′′ = ≠ ′′

-4 -2 2 4

-2

2

4

6

( ) ( )( ) ( )

− = − ⇔ − = −⋅ ⋅

⇔ ⋅ − = ⋅ −

⇔ ⋅ − = − ⋅ −

Page 14: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

-4 -2 2 4

-2

2

= − −

= −

= − − −

= − − − − −

→∞− − − − −

→∞− +

→ ∞

Page 15: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= ⋅ =

( )( )

= − = − = − = −⋅⋅⋅

−= − = − ⋅ = − ⋅

−−= + = −

Page 16: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−= + ⋅ ∈ ∈

Page 17: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

=

− = − + ⋅= + ⋅= + ⋅

⋅= + ⋅ ==

−= + ⋅

Page 18: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−= +

− − − − −= + ⋅ − − = + ⋅ ∈

− − − − −

− − −+ ⋅ = + ⋅

⋅ = −⋅ − ⋅ =⋅ − ⋅ = −

⋅ = −⋅ = −

Page 19: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

= = −

− − −+ ⋅ = + ⋅

− − ⋅ = −+ ⋅ = + ⋅

− + ⋅ = + ⋅

= = −

α

= ≈ ≈ °⋅

α

⋅ + ⋅ =

Page 20: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−= − ⋅ ⇔ = − ⋅

+− = ⋅ +

Page 21: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

∈ +

=− +

=

Page 22: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

===

= = =

=−⋅− −

− −− = − −

− −

+ − −⋅ + ⋅ − − =

− −

⋅ ⋅ − + − ⋅ =− − = −

⋅ ⋅ + − ⋅ − ⋅ =+ =

− − − − − −⇔

− −

=→

−→

⋅−=

⋅ ⋅ = =

Page 23: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

=→→

⋅−=

− −− − ⋅ = =− −

= =→→

⋅−=

→−

→⋅−=

− −− − ⋅ = =− −

− − =− + − =− − + =

⇔ =⇔ =⇔ =

= =

= =

→→⋅−= =

− −− = − −

− − − +

→− −

− − ⋅ = − − =− +− − − +

Page 24: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

→→⋅=

[ ] ⋅ − − =− +

= − − +

′ = − +

′′ = −

′ ′′= ∧ <

( ) ( )

− + =

= ∨ = − ∉′′= <

−=

= − =

Page 25: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

Page 26: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

•μ

Page 27: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

μ

Page 28: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

−= = ⋅ ⋅ −

= = ≈

= = ⋅ ≈

= = ⋅ ⋅ ≈

= = ⋅ ⋅ ≈

= = ⋅ ⋅ ≈

≤ ≈

> = − ≤ ≈ − =

Page 29: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

μ ⋅ ⋅

= ≈ ⋅μμ

= ≈ − = ≈

>

> = − ≤ ≈ − ≈

∞ ∞ ∞

= = =

⋅ ⋅ − ⋅ = ⋅ ⋅ ⋅ − ⋅ ⋅μ μ μμ μ μ

[ ]≈ ⋅ − − ⋅ − [ ]− ⋅ =

Page 30: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

Page 31: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

⋅ + ⋅ + ⋅ =

≤ ≤

−= = −

= ⋅ ⋅ −σ σ = ⋅ ⋅ = >

− −≤ ≤ ≈ Φ − Φ = ⋅ Φ −

≈ ⋅ Φ − ≈ ⋅ − = ≈

=

μ = σ = ⋅ ⋅ ≈

μ σ≤ + ⋅ ≈≤ ≈ > ≈

Page 32: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

≤≤ =μ σ = >

( )− +≤ ≈ Φ ≈ Φ ≈

[ ]

⋅ −− =

⋅ −

− ≤ ≥

Page 33: PQDVLHQ %HK|UGH I U %LOGXQJ XQG 6SRUW =ZHLWWHUPLQ ...groolfs.de/Aufgaben/HH_LK_W_N_2005.pdf · )uhlh xqg +dqvhvwdgw +dpexuj :luwvfkdiwvj\pqdvlhq %hk|ugh i u %logxqj xqg 6sruw =zhlwwhuplq

⋅ − ⋅ − ≈

− ≤ ≈

⋅ −− = ⋅