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IXQNFLMH��� � � � � �� � � � � � � � �� � � � � �� � � � � � � � �� � � � � 0 � � � � � � � �� � � � � 1 � � � � � � � �� � � � � 2 � � � � � � � ��� �� �� �� �� 3 �� �� �� �� �� �� �� ��� 041 2 36543 2 17098;: � �� � � � � 3 � � � � � � � �� � � � � 2 � � � � � � � �� � � � � 1 � � � � � � � �� � � � � �� � � � � � � � �� � � � � �� � � � � � � � ��

•� 'RPHQD�IXQNFLMH�MH�VNXS�5���•� 2YD�IXQNFLMD�LPD�WUL�QXOWRþNH�� 3,1,3 =−=−= xxx ��•� )XQNFLMD�LPD�GYD�ORNDOQD�HNVWUHPD�� maxy ����]D��[� ������ miny � �����]D��[� ����•� 1D LPR�LQWHUYDOH�UDVWD��RGQRVQR�SDGD�IXQNFLMH��

NDGD�[� UDVWH�RG�� �∞� �GR�� ����IXQNFLMD�UDVWH�RG� � �∞� �GR� ����NDGD�[� UDVWH�RG� � ��� �GR�����IXQNFLMD�SDGD�RG�����GR������NDGD�[�UDVWH�RG�����GR���∞��IXQNFLMD�UDVWH�RG������GR��∞��

•� )XQNFLMD� SRSULPD� SR]LWLYQH� YULMHGQRVWL� �LOL 0)( >xf �� QD� VNXSX� +∞∪−− ,31,3 �� D�QHJDWLYQH�YULMHGQRVWL��LOL�� 0)( <xf ��QD�VNXSX�� 3,13, −∪−∞− ��

•� )XQNFLMD�LPD�VOLMHGHüH�JUDQLþQH�YULMHGQRVWL��∞=

∞→)(lim xf

x���]DLVWD��YLGLPR�GD�NDGD�[�WHåL�SUHPD�∞�� )(xf �WDNR HU�WHåL�SUHPD�∞��

−∞=−∞→

)(lim xfx

��� 0)(lim3

=−→

xfx

��� 1)(lim0

−=→

xfx

��� 3)(lim5

=→

xfx

��� 2)(lim4

−=−→

xfx

��LWG������

Page 58: Efzgforum.net Matematika Pred

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��� *5$1,ý1$�95,-('1267��1(35(.,187267��$6,03727(��

'D�EL�REMDVQLOL�SRMDP�JUDQLþQH�YULMHGQRVWL�LOL�OLPHVD�IXQNFLMH�� )(xfy = �X�QHNRM�WRþNL��[ < ��QDFUWDMPR�JUDI�IXQNFLMH�

=≠

=2,1

2,)(

2

x

xxxf ������

�9LGLPR��NDGD� VH�QH]DYLVQD�YDULMDEOD�[� QD�ELOR�NRML�QDþLQ�SULEOLåDYD� WRþNL� �[� ����YULMHGQRVW�IXQNFLMH�� )(xf �SULEOLåDYDMX�VH�EURMX����'DNOH��� 1→x ��⇒�� 1)( →xf , pa pišemo 1)(lim

1=

→xf

x�L�

NDåHPR�GD�RYD�IXQNFLMD�X�WRþNL��[� ����LPD�JUDQLþQX�YULMHGQRVW����$QDORJQR�YULMHGL����

1−→x ��⇒�� 1)( →xf ���WM��� 1)(lim1

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xfx

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2=

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x��

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=→

xfx

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'DNOH��DNR�YULMHGL��� )( 0xx te� i→ �⇒�� ))(( Axf → ��RQGD�VH�NDåH�GD�IXQNFLMD�� )(xf �X�WRþNL���[� �[ = �LPD�JUDQLþQX�YULMHdnost (ili limes) A, i piše Axf

xx=

→)(lim

0

���'HILQLFLMD���1HND�MH�IXQNFLMD�� )(xf �GHILQLUDQD�X�VYLP�WRþNDPD�QHNH�RNROLQH�WRþNH��[ = ��RVLP�PRåGD�X� VDPRM� WRþNL� �[ = ��5HüL�üHPR�GD� MH� �$∈5� � �JUDQLþQD�YULMHGQRVW� LOL� OLPHV� IXQNFLMH�

)(xf � X� WRþNL� � [ = � � L� SLVDWL� � Axfxx

=→

)(lim0

�� DNR� ]D� � 0>∀ε � � 0>∃δ � � WDNDY� GD� YULMHGL��ε<− Axf )( ��þLP�MH�� δ<− 0xx ��L�� 0xx ≠ ��

Page 59: Efzgforum.net Matematika Pred

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1D� VOLþDQ�QDþLQ�GHILQLUD� VH� L�JUDQLþQD�YULMHGQRVW�V� OLMHYD (piše se )(lim0

0

xfxx −→

�� � L� �V�GHVQD��(piše se )(lim

00

xfxx +→

���(YR�GYD�SULPMHUD��

� � > � � �� � ? � � �

� � @ � � ��� �� A �� �� ��@ A6B4A @ ?

� � A � � �

=

=

+

3)(lim

1)(lim

0

0

1

1

xf

xf

x

x ��

%XGXüL�GD�VX�RYH�GYLMH�JUDQLþQH�YULMHGQRVWL�UD]OLþLWH��RYD�IXQNFLMD�QHPD�JUDQLþQX��YULMHGQRVW�X�WRþNL��[� �����

� > � � � �� ? � � � �� @ � � � ���A �� �� �� ��

A B AC@9?7>

� A � � � ��

−∞=

+∞=

+

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)(lim

0

0

3

3

xf

xf

x

x ��8�RYRP�VOXþDMX�NDåHPR�GD�IXQNFLMD�LPD�YHUWLNDOQX�DVLPSWRWX�X�WRþNL�[� ����

�'HILQLFLMD�� � )XQNFLMD� )(xf � MH� � QHSUHNLQXWD� X� WRþNL� � [ D � � DNR� X� WRM� WRþNL� LPD� JUDQLþQX�YULMHGQRVW�L�DNR�MH�WD�JUDQLþQD�YULMHGQRVW�MHGQDND�� )( 0xf ��� )()(lim 0

0

xfxfxx

=→

��)XQNFLMD�� )(xf �MH��QHSUHNLQXWD�DNR�MH�QHSUHNLQXWD�X�VYDNRM�WRþNL�VYRMH�GRPHQH���'UXJLP�ULMHþLPD��IXQNFLMD�� )(xf �MH�QHSUHNLQXWD�X�WRþNL��[ E ��DNR�VH�RUGLQDWH�� )(xf �X�GRYROMQR�PDORM�RNROLQL� WRþNH� �[ E � � UD]OLNXMX�SR�YROML�PDOR�RG� � )( 0xf �� LOL� DNR� MH�QMHQ�JUDI� VXYLVDR�� WM��SRYH]DQ�X�RNROLQL�WRþNH��[ E ��

Page 60: Efzgforum.net Matematika Pred

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$NR� VH� YUDWLPR� QD� SULMH� SURPDWUDQX� IXQNFLMX� �

=≠

=2,1

2,)(

2

x

xxxf � �� � YLGLPR� GD� YULMHGL��

4)(lim2

=→

xfx

��� 1)2( =f ��WM��� )2()(lim2

fxfx

≠→

��SD�WR�]QDþL�GD�RQD�QLMH�QHSUHNLGQD�X�WRþNL�[� ����2QD�X�WRM�WRþNL�LPD�SUHNLG��ãWR�VH�RGPDK�YLGL�L]�QMHQRJ�JUDID���3ULPMHU����

� � F � � � � �� � G � � � � �� � H � � � � ��� �� I �� �� � � ��H I6J4I H G7F9K

� � I � � � � ��

� ⇒

=

=→

1)3(

1)(lim3

f

xfx � ⇒=

→)3()(lim

3fxf

x�RYD�IXQNFLMD�MH�QHSUHNLQXWD�X�� 3=x �

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xx

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xx

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32

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x

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x

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Page 61: Efzgforum.net Matematika Pred

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'HILQLFLMD���3UHWSRVWDYLPR�GD�VH�JUDI�QHNH�IXQNFLMH�SURWHåH�X�EHVNRQDþQRVW��WM��GD�XGDOMHQRVW�WRþDND� WRJ� JUDID� RG� LVKRGLãWD� UDVWH� X� EHVNRQDþQRVW�� $VLPSWRWD� WH� IXQNFLMH� MH� SUDYDF� VD�VYRMVWYRP�GD�XGDOMHQRVW�WRþDND�WRJ�SUDYFD�RG�WRþDND�JUDID�IXQNFLMH�WHåL�N�QXOL�NDGD�WH�WRþNH�WHåH�X�EHVNRQDþQRVW���3UHPD�SRORåDMX�RE]LURP�QD�NRRUGLQDWQL�VXVWDY�DVLPSWRWH�GLMHOLPR�QD�YHUWLNDOQH��KRUL]RQWDOQH�L�NRVH���9HUWLNDOQD� DVLPSWRWD� SDUDOHOQD� MH� V� RVL� \�� 9ULMHGQRVW� IXQNFLMH )(xf � X� WRþNL� � [� � [ L MH�EHVNRQDþQR�YHOLND��SD�YULMHGL��SUDYDF��[� �[ M �MH�YHUWLNDOQD�DVLPSWRWD��IXQNFLMH�� )(xfy = DNR�MH�� ±∞=)( 0xf ���]DWR� )(xf X��[ L QLMH�GHILQLUDQD���

�� � N � � � �� � O � � � �� � P � � � ��� �� Q �� �� � ��

P Q6R4Q P O

� � Q � � � ��1SU��IXQNFLMD��

45

)(2 −

=x

xxf ��LPD�GYLMH�YHUWLNDOQH�DVLPSWRWH��� 2=x ��L��� 2−=x ��

�7HåL� OL� IXQNFLMD� )(xf � RGUH HQRM� JUDQLþQRM� YULMHGQRVWL� � \ L NDGD� � ∞→x �� WM�� DNR� SRVWRML��

∞<=∞→ 0)(lim yxf

x��RQGD�MH�SUDYDF��\� �\ S �KRUL]RQWDOQD�DVLPSWRWD��IXQNFLMH� )(xf ��

�� � N � � �� �

O � � �� � P � � ��� �� Q �� �� ��P Q6R4Q P O� � Q � � �

�1SU���IXQNFLMD��

237

)(+

=x

xxf ��LPD�KRUL]RQWDOQX�DVLPSWRWX���

37=y ��

[ [ T

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Page 62: Efzgforum.net Matematika Pred

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$NR�]D�QHNX�IXQNFLMX�YULMHGL )(xf YULMHGL�� ∞=∞→

)(lim xfx

��WDGD�RQD�PRåGD�LPD�NRVX�DVLPSWRWX��

lkxy += ��$NR�MH�LPD��RQGD�MH��xxf

kx

)(lim

∞→= ���� ))((lim kxxfl

x−=

∞→��

�� � N � � �� � O � � �� � P � � ��� �� Q �� �� ��P Q6R4Q P O� � Q � � �

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\ N[�O�

Page 63: Efzgforum.net Matematika Pred

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��� (/(0(17$51(�)81.&,-(��

7R�VX��QDMMHGQRVWDYQLMH��IXQNFLMH� L]�NRMLK�SRPRüX�RSHUDFLMD�]EUDMDQMD��PQRåHQMD��GLMHOMHQMD��SRWHQFLUDQMD� L� SRPRüX� NRPSR]LFLMH� IXQNFLMD� QDVWDMX� VYH� RVWDOH� IXQNFLMH�� 1SU�� 3

1 )( xxf = ���12)(2 += xxf �� � xxf sin)(3 = � � VX� HOHPHQWDUQH� IXQNFLMH�� D� � )12sin()( += xxg � � L��

)12(sin)( 3 += xxh � � QLVX� �YULMHGL�� ))(()( 23 xffxg = �� � ))(()( 231 xfffxh = ��� 6DGD� üHPR� GDWL�SUHJOHG�HOHPHQWDUQLK�IXQNFLMD����

������32/,120,��

)XQNFLMD� � :f 5�→5� �GHILQLUDQD� IRUPXORP� � 011

1)( axaxaxaxP nn

nnn ++++= −

− " �� JGMH� MH��Q∈1�� ia ∈5��L�= 0, 1, 2, … , n) i na ≠���]RYH�VH��SROLQRP�Q�WRJ�VWXSQMD�LOL�FLMHOD�UDFLRQDOQD�IXQNFLMD��3ROLQRPL� VX�QDMYDåQLMH� L�QDMMHGQRVWDYQLMH� IXQNFLMH��D� VYH�RVWDOH� IXQNFLMH�PRJX� VH�DSURNVLPLUDWL�SROLQRPRP���1SU����

!++++≈!3!2!1

132 xxx

e x ����

�� !+−+−≈!7!5!3

sin753 xxx

xx �3ROLQRPL�VX�QHSUHNLQXWH�IXQNFLMH���3RVHEQR��SROLQRP� � 0)( =xf � ]RYH� VH�QXOSROLQRP�� D�SROLQRP� � 0)( 0 ≠= axf � MH�NRQVWDQWD� L�QMHJRY�JUDI�MH�SUDYDF�SDUDOHODQ�V�RVL�[��

�3ROLQRP�REOLND�� nxxf =)( ��]RYH�VH�SULURGQD�SRWHQFLMD��WX�MH�GDNOH�� ,1=na �

0011 ====− aaan ! �����

I�[� D V

Page 64: Efzgforum.net Matematika Pred

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1D�VOLMHGHüLP�VOLNDPD�SULND]DQL�VX�JUDIRYL�QHNLK�SULURGQLK�SRWHQFLMD��

���� ���� ���� �� ������ xxf =)( � � ����� 2)( xxf = � � ���� 3)( xxf = � � ���� 4)( xxf = �9LGLPR� GD� VX� 2)( xxf = � L� � 4)( xxf = � � SDUQH� IXQNFLMH� L� GD� SRSULPDMX� VDPR� QHQHJDWLYQH�YULMHGQRVWL��D�GD�VX�� xxf =)( �L�� 3)( xxf = ��QHSDUQH�IXQNFLMH�L�UDVWXüH�IXQNFLMH���7HRUHP�R�QXOWRþNDPD�SROLQRPD���3ROLQRP�Q�WRJ�VWXSQMD�� 01

11)( axaxaxaxP n

nn

nn ++++= −− " �LPD��Q��QXOWRþDND�LOL�NRULMHQD�X�

VNXSX�&��WM��SRVWRML��Q��EURMHYD�� ∈nxxx ,,, 21 ! &��WDNYLK�GD�MH�� 0)()()( 21 ==== nxfxfxf " ���3ROLQRP�� )(xf ��PRåHPR�]DSLVDWL�X�REOLNX�� )())(()( 21 nn xxxxxxaxf −⋅⋅−−= ! ��1HNL�RG�WLK�EURMHYD�� ix ��VX�UHDOQL��D�QHNL�QLVX�UHDOQL��$NR�VX�WL�NRULMHQL�� ix ��UHDOQL�EURMHYL��RQL�VX�QXOWRþNH�SROLQRPD��7R�]QDþL�GD�SROLQRP�Q�tog stupnja ima najviše Q��QXOWRþNL���3ULPMHUL��

•� 3ROLQRP 7272)( 23 −−+= xxxxf � � LPD� VYD� WUL� UHDOQD� NRULMHQD�� WM�� 11 =x �� 12 −=x ��

27

3 −=x � �WR� VX� UMHãHQMD� MHGQDGåEH� 07272 23 =−−+ xxx ��� SD� MH��

)27

)(1)(1(2)( ++−= xxxxf �

•� 3ROLQRP�� 1)( 2 += xxf ��QHPD��QLWL�MHGDQ�UHDODQ�NRULMHQ���7HRUHP�R�GLMHOMHQMX�SROLQRPD���%LOR�NRMD�GYD�SROLQRPD�� )(xf �L�� )(xg �RGUH XMX�MHGLQVWYHQH�SROLQRPH�� )(xs �L�� )(xr �WDNYH�GD�MH�� )()()()( xrxgxsxf +⋅= ��RGQRVQR���

)()(

)()()(

xgxr

xsxgxf += ��

JGMH� MH� � VWXSDQM� )(xr �� VWXSDQM� )(xg �� 3ROLQRP� )(xs � ]RYH� VH� NYRFLMHQW�� D� SROLQRP� )(xr �RVWDWDN�GLMHOMHQMD��$NR�MH�� )(xr ����NDåHPR�GD�� )(xg �GLMHOL�� )(xf ��

Page 65: Efzgforum.net Matematika Pred

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2YDM�WHRUHP�NRULVWLPR�X�VOXþDMX�NDGD�MH��VW )(xf ≥ VW� )(xg �L�X�WRP�VOXþDMX�MH�VW� )(xs �VW )(xf ��VW� )(xg ���3ULPMHU��1HND�MH�� 132)( 234 −+−+= xxxxxf ��� xxxg 2)( 2 −= ��SD�QD LPR� )(xs L� )(xr ���� ( ):132 234 −+−+ xxxx ( ) 7522 22 ++=− xxxx �� � 34 42 xx − �

�����������������������23

23

105

135

xx

xxx

−+− �

� � �xx

xx

147

172

2

−+ �

� � � ���������� 115 −x ��'DNOH��NYRFLMHQW�MH�� )(xs � 752 2 ++ xx ��D�RVWDWDN�� )(xr � 115 −x ���=DLVWD�YULMHGL�� )()()()( xrxgxsxf +⋅= �L��VW� )(xr ��VW )(xg ���

132 234 −+−+ xxxx � ( )( ) ( )1152752 22 −+−++ xxxxx �LOL�

xxx

xxxx

xxxx2115

7522

1322

22

234

−−+++=

−−+−+ ���

�3ULPMHUL��

•� 32333232344812 )1()1()1()1()1()1(133)( ++−=+−=−=−+−= xxxxxxxxxxf �•� 223 )2)(1(43)( −+=+−= xxxxxf �•� )1)(2(2)( 223 ++−=−−−= xxxxxxxf �

���3ROLQRP�SUYRJ�VWXSQMD�MH�IXQNFLMD�� :f 5→5��]DGDQD�IRUPXORP�� 01)( axaxf += �� 01 ≠a ��LOL�

lkxxf +=)( �� 0≠k ��2YD�IXQNFLMD�]RYH�VH�OLQHDUQD�IXQNFLMD��1MHQ�JUDI�MH�SUDYDF��

Page 66: Efzgforum.net Matematika Pred

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��

kx

llkxtg =−+=α �⇒� αtgk = ��SD�VH��N��]RYH�NRHILFLMHQW�VPMHUD�SUDYFD��

$NR�MH�� 0>k �⇒��

2,0πα �⇒�IXQNFLMD�MH�UDVWXüD��

$NR�MH�� 0<k �⇒��

∈ ππα ,

2�⇒�IXQNFLMD�MH�SDGDMXüD��

/DNR�VH�YLGL�GD�MH�OLQHDUQD�IXQNFLMD�ELMHNFLMD��MHU�YULMHGL�� 2121 )()( xxxfxf =⇒= �L�� ∈∀ 0y 5��∈∃ 0x 5�� 00 )( yxf = ���SD�LPD�LQYHU]QX�IXQNFLMX�NRMD�MH�RSHW�OLQHDUQD�IXQNFLMD��

�3ULPMHU���1D LPR�LQYHU]QX�IXQNFLMX�IXQNFLMH�� 32)( −= xxf ��

xxff =− ))(( 1D � � ⇒� � xxff =− ))(( 1 � � ⇒� � xxfxff =−⋅= −− 3)(2))(( 11 � � ⇒��

23

21

)(1 +=− xxf ���*UDIRYL�WLK�IXQNFLMD�VX���

�� � � � ������� xy = � ��������� 32 −= xy ��9LGLPR� GD� VX� JUDIRYL� WLK� IXQNFLMD� VLPHWULþQL� RE]LURP� QD� SUDYDF� � xxf =)( �� 7R� YULMHGL� L�RSüHQLWR��JUDIRYL�IXQNFLMD� )(xf �L�� )(1 xf − �VLPHWULþQL�VX�RE]LURP�QD�SUDYDF�� xxf =)( ��

[��O��N�

7�[�N[�O��O�

α�

α�

23

21 += xy �

Page 67: Efzgforum.net Matematika Pred

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QD]LYQLNX�GLIHUHQFLMD�LOL�SULUDVW�DUJXPHQWD��$NR�IXQNFLMD� )(xf �LPD�GHULYDFLMX�X�VYDNRM�WRþNL�VYRMH�GRPHQH��WM��DNR�SRVWRML�

)(xf ′ �dx

xdf )( � �x

xfxxfx ∆

−∆+→∆

)()(lim

0� �

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h

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0

−+→

���∀[∈'�I����WDGD�VH�NDåH�GD�MH�I�GHULYDELOQD�IXQNFLMD��D�IXQNFLMD� )(xfy ′= �]RYH�VH�GHULYDFLMD�IXQNFLMH�I��

V�

W�7�[ X �∆[�I��[ X �∆[��

7 X �[ X �I��[ X ���

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3RMDP�GHULYDFLMH� IXQNFLMH� XYHOL� VX�1HZWRQ� L� /HLEQL]� X� GUXJRM� SRORYLQL� ���� VWROMHüD� GRN� VX�Uješavali problem brzine, odnosno problem tangente. Od tada je derivacija jedaQ� RG�QDMYDåQLMLK�L�QDMYLãH�SULPMHQMLYDQLK�PDWHPDWLþNLK�SRMPRYD���1D LPR�MHGQDGåEX�WDQJHQWH�X�WRþNL� 0x �QD�JUDI�IXQNFLMH� )(xf ��3UDYDF�NUR]�WRþNH�7 Y L�7�]RYH�VH�VHNDQWD��D�JUDQLþQL�SRORåDM�WRJ�SUDYFD�NDG� 0TT te� i→ �]RYH�VH�WDQJHQWD��YLGL�VOLNX���$NR�NULYXOMD� )(xf �LPD�WDQJHQWX�� )( 00 xxkyy −=− �X�WRþNL� 0x ��RQGD�YULMHGL��

=== αα tgtgk lim0 xf

x ∆∆

→∆ 0lim � �

xxfxxf

x ∆−∆+

→∆

)()(lim

0� � )( 0xf ′ �� SD� MH� WUDåHQD� MHGQDGåED�

WDQJHQWH�� � )()( 000 xxxfyy −⋅′=− �� ,]� WRJD� VOLMHGL� JHRPHWULMVNR� ]QDþHQMH� GHULYDFLMH��GHULYDFLMD� X� WRþNL� 0x � IXQNFLMH� )(xf � MH� NRHILFLMHQW� VPMHUD� WDQJHQWH� X� WRþNL� 0x � QD� NULYXOMX�

)(xfy = ��$QDORJQR� GR� GHULYDFLMH� GROD]LPR� NRG� RGUH LYDQMD� EU]LQH� JLEDQMD� QHNRJ� WLMHOD�� 1HND� MH�

)(tss = �SULMH HQL�SXW�L]UDåHQ�NDR�IXQNFLMD�YUHPHQD���7DGD�MH�SURVMHþQD�EU]LQD�

ttstts

v∆

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tsttstv

t ∆−∆+

=→∆

)()(lim)( 00

00 ��

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tvttvta

t ∆−∆+=

→∆

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00 ���3ULPMHU�����1D LPR�MHGQDGåEX�WDQJHQWH�QD�NULYXOMX� 1)( 2 −= xxf �X�WRþNL�� 0x ����� W… )()( 000 xxxfyy −⋅′=− �� 1)( 2 −= xxf ��⇒�� xxf 2)( =′ ���YLGL�SULPMHU������ 0x ����⇒�� 011)( 0 =−=xf ��� 212)( 0 =⋅=′ xf �

⇒� W… )1(20 −⋅=− xy ������� 22 −= xy ��

Page 79: Efzgforum.net Matematika Pred

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3ULPMHU�����1D LPR�GHULYDFLMX�IXQNFLMH�� 1)( 2 −= xxf ��)(xf ′ � �

hxfhxf

h

)()(lim

0

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� � =−−−+→ h

xhxh

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22

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→ hxhxhx

h

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222

0�

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xxf

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0

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+→ h

xhxh

11

lim0

=++−

→ hhxxhxx

h

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lim0 20

1)(

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+−

→�

�1D�VOLþDQ�QDþLQ�GROD]L�VH�GR�VOLMHGHüH�WDEOLFH�GHULYDFLMD�HOHPHQWDUQLK�IXQNFLMD�L�RVQRYQLK�SUDYLOD�GHULYLUDQMD���

��� == Cy NRQVWDQWD� ⇒� 0=′y ���� ∈= αα ,xy 5�� ⇒� 1−⋅=′ αα xy � �SRVHEQR�� xy = ��⇒�� 1=′y ����� xay = �� � ⇒� aay x ln⋅=′ � �SRVHEQR�� xey = ��⇒�� xey =′ ����� xy alog= � � ⇒�

axy

ln11 ⋅=′ ����SRVHEQR�� xy ln= ��⇒��

xy

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xy

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1

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1

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1x

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Page 80: Efzgforum.net Matematika Pred

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3UDYLOD����� )()( xgxfy ±= � � ⇒� )()( xgxfy ′±′=′ ���� )()( xgxfy ⋅= � � ⇒� )()()()( xgxfxgxfy ′⋅+⋅′=′ �

�SRVHEQR��� )(xfCy ⋅= ��⇒�� )(xfCy ′⋅=′ �����

)()(

xgxf

y = � � � ⇒�)(

)()()()(2 xg

xgxfxgxfy

′⋅−⋅′=′ �

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dfdg

xfxfgy ⋅=′⋅′=′ )())(( ��3ULPMHUL��

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3 2

32

131

31

3

3

131

31

xxxyxyxy

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72621117133 627 +−=′⇒++−= xxyxxxy �

xxxxyxxy

1ln2ln 22 ⋅+⋅=′⇒⋅= �

4

2

2

2)5(15x

xxxy

xx

y⋅+−⋅=′⇒+= �

)16()13(4)13( 3242 +⋅++=′⇒++= xxxyxxy �xxyxy 6)13sin()13cos( 22 ⋅+−=′⇒+= �

)110()5(4)5sin()5cos( 324242 +⋅+⋅+−=′⇒+= xxxxxyxxy �11)511cos()511(sin7)511(sin 67 ⋅+⋅+=′⇒+= xxyxy �

10ln1010 ⋅=′⇒= xx yy �

21

7arcsin7

xyxy

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)1(2

1121 2

1

xxx

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Page 81: Efzgforum.net Matematika Pred

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2. VIŠE DERIVACIJE I EKSTREMI��

$NR� IXQNFLMD� )(xf � LPD� GHULYDFLMX�� RQGD� MH� QMHQD� GHULYDFLMD� )(xf ′ � QRYD� IXQNFLMD�� $NR� L�IXQNFLMD� )(xf ′ � LPD� VYRMX� GHULYDFLMX�� RQGD� VH� NDåH� GD� MH� WD� GHULYDFLMD� GUXJD� GHULYDFLMD�SROD]QH� IXQNFLMH� )(xf �� 'DNOH�� ))(()( ′′=′′ xfxf �� RGQRVQR� �

hxfhxf

xfh

)()(lim)(

0

′−+′=′′

→��

$QDORJQR�VH�GHILQLUD�WUHüD�GHULYDFLMD�� ))(()( ′′′=′′′ xfxf ��LWG���3ULPMHUL����

2345 60)(20)(5)()( xxfxxfxxfxxf =′′′⇒=′′⇒=′⇒= ��LWG��xxfxxfxxfxxfxxf IV sin)(cos)(sin)(cos)(sin)( =⇒−=′′′⇒−=′′⇒=′⇒= �

xxx exfexfexf =′′⇒=′⇒= )()()( ��LWG���7HRUHP� �XYMHW�PRQRWRQRVWL��� �$NR�]D�GHULYDFLMX� IXQNFLMH� I�]D� VYDNL�[� L]�QHNRJ� LQWHUYDOD� ,�YULMHGL� 0)( >′ xf �� RQGD� MH� IXQNFLMD� I� VWURJR� UDVWXüD� QD� ,�� D� DNR� YULMHGL� 0)( <′ xf �� RQGD� MH�IXQNFLMD�I�VWURJR�SDGDMXüD�QD�,��'RND]��� 6OLMHGL�L]�GHILQLFLMH�GHULYDFLMH��LOL�L]�JHRPHWULMVNRJ�]QDþHQMD�GHULYDFLMH���

��⇒� � 0)()()()( 4321 =′=′=′=′ xfxfxfxf � �MHU� X� WLP� WRþNDPD� UDVW� IXQNFLMH� SUHOD]L� X� SDG� LOL�REUDWQR��RGQRVQR�X�WLP�WRþNDPD�IXQNFLMD�LPD�PDNVLPXP�LOL�PLQLPXP��WM��HNVWUHP����)XQNFLMD� I� LPD� X� WRþNL� 0x � PDNVLPXP� DNR� SRVWRML� RNROLQD� WRþNH� 0x � WDNYD� GD� YULMHGL��

)()( 0xfxf < �]D�VYDNL�� 0xx ≠ �L]�WH�RNROLQH��RGQRVQR�PLQLPXP�DNR�SRVWRML�RNROLQD�WRþNH� 0x �WDNYD�GD�YULMHGL�� )()( 0xfxf > �]D�VYDNL�� 0xx ≠ �L]�WH�RNROLQH���

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Page 82: Efzgforum.net Matematika Pred

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9ULMHGQRVW� IXQNFLMH� I� X� WRþNL� � 0x �� WM�� )( 0xf �� WDGD� VH� ]RYH� PDNVLPXP� RGQRVQR� PLQLPXP�IXQNFLMH�LOL�NUDWNR�HNVWUHP�IXQNFLMH��6OLMHGHüL�WHRUHP�JRYRUL�R�WRPH�NDNR�VH�SURQDOD]H�HNVWUHPL���7HRUHP�R�HNVWUHPLPD� IXQNFLMH�� �$NR� IXQNFLMD� I� LPD�X� WRþNL�HNVWUHP��RQGD� MH� 0)( 0 =′ xf ��$NR�MH�� 0)( 0 <′′ xf �RQGD�MH�WDM�HNVWUHP�PDNVLPXP��D�DNR�MH� 0)( 0 >′′ xf �RQGD�MH�PLQLPXP��$NR�MH�SDN� 0)( 0 =′′ xf ��RQGD�WUHED�WUDåLWL�YLãH�GHULYDFLMH�GRN�VH�QH�GR H�GR�GHULYDFLMH�NRMD�MH�X� 0x �UD]OLþLWD�RG�QXOH��$NR�MH�RQD�SDUQRJ�UHGD��X�WRM�WRþNL�MH�HNVWUHP��L�WR�PDNVLPXP�DNR�MH�RQD�PDQMD�RG�QXOH��D�PLQLPXP�DNR�MH�YHüD�RG�QXOH���D�DNR�MH�RQD�QHSDUQRJ�UHGD��X�WRM�WRþNL�QLMH�HNVWUHP��,]� RYRJ� WHRUHPD� VOLMHGL� GD� MH� QXåDQ� XYMHW� ]D� HNVWUHP� 0)( =′ xf �� WM�� X� WRþNL� HNVWUHPD� SUYD�GHULYDFLMD�PRUD�ELWL�MHGQDND�QXOL��WR�MH�MDVQR�L�]ERJ�JHRPHWULMVNRJ�]QDþHQMD�GHULYDFLMH��X�WRþNL�HNVWUHPD� WDQJHQWD� QD� NULYXOMX� SDUDOHOQD� MH� V� RVL� [��� 3UHPD� WRPH�� PRJXüL� HNVWUHPL� VX� X�WRþNDPD� NRMD� VX� UMHãHQMD� MHGQDGåEH� 0)( =′ xf �� 5MHãHQMD� WH� MHGQDGåEH� ]RYX� VH� VWDFLRQDUQH�WRþNH��D�GD�OL�MH�X�QMLPD�]DLVWD�HNVWUHP�WUHED�LVSLWDWL�SRPRüX�GUXJH�GHULYDFLMH�LOL�SRPRüX�MRã�viših derivacija.��3ULPMHU�����2GUHGL�HNVWUHPH�IXQNFLMH�� xxxxf 36152)( 23 −−= �� 1,6065036306)( 21

22 −==⇒=−−⇒=−−=′ xxxxxxxf ��VWDFLRQDUQH�WRþNH��� 3012)( −=′′ xxf �

⇒� � 04230612)6( >=−⋅=′′f � �⇒� � X� 61 =x � MH� PLQLPXP�� L� YULMHGQRVW� WRJ����PLQLPXPD�MH�� 32463661562)6( 23 −=⋅−⋅−⋅=f �⇒�� 04230)1(12)1( <−=−−⋅=−′′f ��⇒��X� 12 −=x �MH�PDNVLPXP��L�YULMHGQRVW�WRJ�PDNVLPXPD��MH�� 19)1(36)1(15)1(2)1( 23 =−⋅−−⋅−−⋅=−f �

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321

)(−+=

xx

xf �

� 0)32(

5)32(

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222=

−−=

−−−−=

−⋅+−−⋅=′

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xxx

xf �� 2YD�MHGQDGåED�QHPD�UMHãHQMD��SD�IXQNFLMD�QHPD�VWDFLRQDUQX�WRþNX��GDNOH�QL�HNVWUHPD��

Page 83: Efzgforum.net Matematika Pred

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34

)(2 +

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lim)(lim2

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2,12 ±=⇒=− xx ��VX�VWDFLRQDUQH�WRþNH�

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Page 84: Efzgforum.net Matematika Pred

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3ULPMHU����� xxxxf 36152)( 23 −−= �•� 'RPHQD�MH�VNXS�5�•� 1XOWRþNH� 0)36152( 2 =−− xxx �⇒ 01 =x �

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Page 85: Efzgforum.net Matematika Pred

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−=

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xf �

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ln)(2

2

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xf �

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Page 86: Efzgforum.net Matematika Pred

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9���,17(*5$/1,�5$ý81���

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)(xf �DNR�YULMHGL�� )()( xfxF =′ ��RGQRVQR�� )()(

xfdx

xdF = ��RGQRVQR�� dxxfxdF )()( = ���3ULPMHUL��

•� 3ULPLWLYQD�IXQNFLMD�IXQNFLMH�� xxf cos)( = ��MH�� xxF sin)( = ���MHU�MH�� xx cos)(sin =′ ��•� 3ULPLWLYQD�IXQNFLMD�IXQNFLMH��

xxf

1)( = ��MH�� xxF ln)( = ���MHU�MH��

xx

1)(ln =′ ��

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31

)( xxF = ���MHU�MH�� 23 )31

( xx =′ ��$NR�MH� )(xF �SULPLWLYQD�IXQNFLMD�RG� )(xf ��WDGD�MH�L�� CxF +)( ��JGMH�MH�� ∈C 5�QHND�NRQVWDQWD��SULPLWLYQD� IXQNFLMD� RG� )(xf �� MHU� YULMHGL�� � ( ) )(0)()( xfxFCxF =+′=′+ �� 2EUDWQR�� DNR� VX��

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Page 87: Efzgforum.net Matematika Pred

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Page 88: Efzgforum.net Matematika Pred

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Page 89: Efzgforum.net Matematika Pred

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Page 91: Efzgforum.net Matematika Pred

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Page 93: Efzgforum.net Matematika Pred

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Page 94: Efzgforum.net Matematika Pred

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n� L� NDåH� VH� GD� GLYHUJLUDMX� X� XåHP� VPLVOX� �LVWR� VH� NDåH� L� DNR� YULMHGL�

−∞=∞→ nn

alim ��� 1L]� MH� GLvergentan u širem smislu� DNR� MH� GLYHUJHQWDQ�� D� QLMH�GLYHUJHQWDQ�X�XåHP�VPLVOX��1SU��QL]RYL�L]�SUHWKRGQLK�SULPMHUD���L���GLYHUJHQWQL�VX�X�širem smislu.�

Page 97: Efzgforum.net Matematika Pred

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'HILQLFLMD���$ULWPHWLþNL�QL]�MH�QL]� )( na �NRMHPX�þODQRYL�LPDMX�VYRMVWYR� daa nn =−+1 ��∀Q∈1��.RQVWDQWD�G�]RYH�VH�GLIHUHQFLMD�DULWPHWLþNRJ�QL]D���=D��Q!��YULMHGL�� =− −1nn aa nn aa −+1 ��RGQRVQR��

211 +− +

= nnn

aaa ��WM��VYDNL�þODQ��RVLP�SUYRJ��

DULWPHWLþNRJ�QL]D�MH�DULWPHWLþND�VUHGLQD�VXVMHGQLK�þODQRYD���,]�RYH�GHILQLFLMH�VOLMHGL�� ,, 121 daaa += ,...,2123 dadaa +=+= dnadaa nn )1(11 −+=+= − �SD�DULWPHWLþNL�QL]�PRåHPR�]DSLVDWL�X�REOLNX�� ,, 11 daa + ,...,21 da + dna )1(1 −+ ,… . Iz toga

GDOMH� VOLMHGL� GD� MH� DULWPHWLþNL�QL]� GLYHUJHQWDQ�X� XåHP� VPLVOX�� +∞=∞→ nn

alim � � �DNR� MH�G�!� ����−∞=

∞→ nnalim ���DNR�MH��G�������

�0RåH�VH�SRND]DWL�GD�MH�]EURM�SUYLK�Q�þODQRYD�DULWPHWLþNRJ�QL]D�MHGQDN��

)(2 11 nnn aan

aas +=++= " �LOL�� [ ]dnan

sn )1(22 1 −+= ��

�3ULPMHUL����

��� �����1, 2, 5, 8, … (tu je 3,41 =−= da ������ �������������18, … (tu je 6,01 −== da ���

�'HILQLFLMD�� �*HRPHWULMVNL�QL]� MH�QL]� )( na �NRMHPX�þODQRYL� LPDMX�VYRMVWYR� � q

aa

n

n =+1 �∀Q∈1��.RQVWDQWD�T��]RYH�VH�NYRFLMHQW�JHRPHWULMVNRJ�QL]D���=D��Q�!����YULMHGL��

n

n

n

n

aa

aa 1

1

+

= ��RGQRVQR�� 11 +− ⋅= nnn aaa ��WM��VYDNL�þODQ�JHRPHWULMVNRJ�QL]D�MH�JHRPHWULMVND�VUHGLQD�VXVMHGQLK�þODQRYD���,]� RYH� GHILQLFLMH� VOLMHGL�� ,, 121 qaaa ⋅= ,...,2

123 qaqaa ⋅=⋅= 111

−− ⋅=⋅= n

nn qaqaa � SD�JHRPHWULMVNL�QL]�PRåHPR�SLVDWL�X�REOLNX��� ,, 11 qaa ,...,2

1 qa ⋅ 11

−⋅ nqa , … ��

Page 98: Efzgforum.net Matematika Pred

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0RåH�VH�SRND]DWL�GD�MH�]EURM�SUYLK�Q�þODQRYD�JHRPHWULMVNRJ�QL]D�MHGQDN��11

1 −−⋅=

qq

asn

n ��L�GD�

MH�JHRPHWULMVNL�QL]�NRQYHUJHQWDQ�]D� 1<q �L�� 1=q ��D�X�RVWDOLP�VOXþDMHYLPD�MH�GLYHUJHQWDQ���3ULPMHUL��

��� ���21 ��

41 ��

81

, … (tu je 21

,11 == qa ��SD�MH�RYDM�QL]�NRQYHUJHQWDQ���

��� ������29 ��

427

, … (tu je 23

,21 == qa ��SD�MH�RYDM�QL]�GLYHUJHQWDQ����1D�NUDMX�QDYHGLPR�QL]�NRML�MH�QDMYDåQLML�]D�PDWHPDWLNX�L�QMHQH�SULPMHQH�X�WHKQLFL��HNRQRPLML��

VWDWLVWLFL�L�GU��7R�MH�QL]�n

n na

+= 11 ��RGQRVQR�QL]��

1

11

1

+ �

2

21

1

+ �

3

31

1

+ ,… ,

n

n

+ 11 ,… �

0RåH� VH� SRND]DWL� GD� MH� WDM� QL]�PRQRWRQR� UDVWXüL� L� GD� MH� RJUD HQ� �GDNOH� YULMHGL� na � 1+na � L��32 <≤ na � ]D� �∀Q∈1��� SD� L]� WH� GYLMH� þLQMHQLFH� VOLMHGL� GD� MH� NRQYHUJHQWDQ�� RGQRVQR�GD� LPD�

JUDQLFX�� 1MHJRYD� JUDQLFD� MH� SR]QDWL� LUDFLRQDOQL� EURM� � ...7182.2=e �� D� WR� ]QDþL� GD� YULMHGL�WHRUHP��� e

n

n

n=

+

∞→

11lim ��

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Page 99: Efzgforum.net Matematika Pred

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����5('29,��

'HILQLFLMD���$NR�MH� )( na �QL]�UHDOQLK�EURMHYD��RQGD�VH�L]UD]�REOLND�

∑∞

=

=+++++1

321k

kn aaaaa "" �]RYH�EHVNRQDþDQ�UHG�LOL�UHG���5HGX�∑

=1kka �PRåH�VH�SULGUXåLWL�QMHJRY�QL]�SDUFLMDOQLK�VXPD� )( ns �þLML�VX�þODQRYL�GHILQLUDQL�QD�

VOMHGHüL�QDþLQ�� nn aaaas ++++= "321 ��WM�� 11 as = �� 212 aas += �� 3213 aaas ++= ��LWG���'HILQLFLMD�� � .DåH� VH� GD� MH� UHG� ∑

=1kka � NRQYHUJHQWDQ� DNR� MH� NRQYHUJHQWDQ� QMHJRY� QL]�

SDUFLMDOQLK�VXPD� )( ns ��*UDQLFD�WRJ�QL]D�]RYH�VH�VXPD�UHGD�L�R]QDþDYD�VD�6��

'DNOH�� Ssnn=

∞→lim �L�X�WRP�VOXþDMX�VH�SLãH�� Sa

kk =∑

=1

��5HG�MH�GLYHUJHQWDQ�DNR�QLMH�NRQYHUJHQWDQ���3ULPMHUL��

��� "" +++++n1

31

21

1 ���

��� "" +++++ −121

41

21

1n ���

��� "" +−+−+−+

n

n 1)1(31

21

1 ������ "" +++++ n321 ������ "" +−+−+− + nn 1)1(321 ���

�.RG�UD]PDWUDQMD�UHGRYD�RVQRYQL�SUREOHP�MH�RGUHGLWL�GD�OL�MH�UHG�NRQYHUJHQWDQ�LOL�GLYHUJHQWDQ�L�X�VOXþDMX�GD�MH�NRQYHUJHQWDQ�L]UDþXQDWL�PX�VXPX�6���9DåDQ�SULPMHU�UHGD�MH�JHRPHWULMVNL�UHG��7R�MH�UHG�REOLND��

++ qaa 11 "+⋅ 21 qa "+⋅+ −1

1nqa �

Page 100: Efzgforum.net Matematika Pred

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,]�SR]QDWRJ�L]UD]D�]D�]EURM�SUYLK�Q�þODQRYD�JHRPHWULMVNRJ�QL]D��11

1 −−⋅=

qq

asn

n ��PRåH�VH�

]DNOMXþLWL�GD�MH�JHRPHWULMVNL�UHG�NRQYHUJHQWDQ�VDPR�X�VOXþDMX�NDGD�MH� 1<q �L�WDGD�PX�MH�VXPD�

qa

sS nn −==

∞→ 1lim 1 ��

�3ULPMHU���3URPRWULPR�UHG� "" +++++ −12

141

21

1n ���2GPDK�VH�YLGL�GD�MH�WR�JHRPHWULMVNL�UHG�

NRG�NRMHJD�MH�21

,11 == qa ��=ERJ� 121

21 <==q �� WDM�UHG�MH�NRQYHUJHQWDQ�L�QMHJRYD�VXPD�MH�

2

21

1

11

1 =−

=−

=q

aS ���

'DNOH�� 281

41

21

1 =++++ " ���3UL� LVSLWLYDQMX� UHGRYD� YHOLNX� SRPRü� GDMX� QDP� GDOMH� QDYHGHQL� WHRUHPL�� RGQRVQR� �NULWHULML��SRPRüX�NRMLK�þHVWR�PRåHPR�LVSLWDWL�SRQDãDQMH�UHGD���7HRUHP��QXåDQ�XYMHW�NRQYHUJHQFLMH���'D�EL�UHG�NRQYHUJLUDR��QXåQR�MH��DOL�L�QH�GRYROMQR��GD�YULMHGL�� 0lim =

∞→ nna ��

�3ULPMHUL����

D�� "" +++++n1

31

21

1 ��

9ULMHGL�� =∞→ nn

alim 01

lim =∞→ nn

��SD�RYDM�UHG�]DGRYROMDYD�QXåDQ�XYMHW�NRQYHUJHQFLMH��D�WR�]QDþL� GD� PRåH� ELWL� NRQYHUJHQWDQ�� DOL� L� QH� PRUD�� 0RåH� VH� SRND]DWL� GD� MH� RYDM� UHG�GLYHUJHQWDQ��WM��GD�YULMHGL� +∞=+++++ ""

n1

31

21

1 ��2YDM�UHG�]RYH�VH�KDUPRQLMVNL�UHG����

Page 101: Efzgforum.net Matematika Pred

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E�� "" +++++ −121

41

21

1n �

9ULMHGL�� =∞→ nn

alim 02

1lim

1=−∞→ nn

��SD�RYDM� UHG�]DGRYROMDYD�QXåDQ�XYMHW�NRQYHUJHQFLMH��D�WR� ]QDþL� GD� PRåH� ELWL� NRQYHUJHQWDQ�� D� PRåH� ELWL� L� GLYHUJHQWDQ�� 1HãWR� UDQLMH� VPR�SRND]DOL�GD�MH�RYDM�UHG�NRQYHUJHQWDQ��7R�MH�JHRPHWULMVNL�UHG�L�VXPD�PX�MH��6� �����

F�� "" ++

++++127

352

31

nn �

9ULMHGL�� � =∞→ nn

alim 021

12lim ≠=

+∞→ nn

n�� D� WR� ]QDþL� GD� RYDM� UHG� QH� ]DGRYROMDYD� QXåDQ�

XYMHW�NRQYHUJHQFLMH��RGQRVQR��RYDM�UHG�MH�GLYHUJHQWDQ���7HRUHP��R�XVSRUH LYDQMX�UHGRYD����1HND�VX�∑

=1nna �L��∑

=1nnb �UHGRYL�V�SR]LWLYQLP�þODQRYLPD�L�

QHND�MH� nn ba ≤ ��∀Q∈1��7DGD�YULMHGL��

D�� DNR�MH�∑∞

=1nnb �NRQYHUJHQWDQ�UHG��RQGD�MH�L�∑

=1nna NRQYHUJHQWDQ�UHG��

E�� DNR�MH�∑∞

=1nna �GLYHUJHQWDQ�UHG��RQGD�MH�L�∑

=1nnb �GLYHUJHQWDQ�UHG��

�3RPRüX� RYRJ� WHRUHPD� GRND]XMX� VH� VOMHGHüD� GYD� WHRUHPD�� RGQRVQR� NULWHULMD� ]D� LVSLWLYDQMH�UHGRYD���7HRUHP� �'$OHPEHUWRY�NULWHULM��� �1HND� MH� ∑

=1nna � UHG�V�SR]LWLYQLP�þODQRYLPD� L�QHND�YULMHGL�

Aa

a

n

n

n=+

∞→

1lim ��$NR�MH��•� $�����UHG�MH�NRQYHUJHQWDQ�•� $!����UHG�MH�GLYHUJHQWDQ�•� $ ����QHPD�RGOXNH��

��

Page 102: Efzgforum.net Matematika Pred

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7HRUHP� �&DXFK\MHY� NULWHULM��� � 1HND� MH� ∑∞

=1nna � UHG� V� SR]LWLYQLP� þODQRYLPD� L� QHND� YULMHGL�

Aannn

=∞→

lim ��$NR�MH��•� $�����UHG�MH�NRQYHUJHQWDQ�•� $!����UHG�MH�GLYHUJHQWDQ�•� $ ����QHPD�RGOXNH��

�3ULPMHUL����

��� ∑∞

=1 !3

n

n

n�

'D�EL�LVSLWDOL�NRQYHUJHQFLMX�RYRJ�UHGD�SULPLMHQLPR�'$OHPEHUWRY�NULWHULM��

=+

∞→n

n

n aa 1lim 0

13

lim

!3

)!1(3

lim

1

=+

=+∞→

+

∞→ nn

nnn

n

n��⇒�� 10 <=A ��SD�MH�RYDM�UHG�NRQYHUJHQWDQ��

���� ∑

= +1 )12(23

nn

n

n�

⇒�� =+

∞→n

n

n aa 1lim

[ ]23

0202

23

32

12

lim23

3212

23

lim

)12(23

1)1(223

lim1

1

=++⋅=

+

+=

++⋅=

+

++∞→∞→

+

+

∞→

n

nnn

n

nnn

n

n

n

n

n�

⇒�� 123 >=A ��SD�MH�RYDM�UHG�GLYHUJHQWDQ��

���� ∑

=1

5

nn

n

n�

%XGXüL� GD� VX� þODQRYL� RYRJ� UHGD� SRWHQFLMH� V� HNVSRQHQWRP�Q�� RYGMH� üHPR� SULPLMHQLWL�&DXFK\MHY�NULWHULM��

05

lim5

limlim ===∞→∞→∞→ nn

an

nn

n

nn

nn��⇒�� 10 <=A ��SD�MH�RYDM�UHG�NRQYHUJHQWDQ��

Page 103: Efzgforum.net Matematika Pred

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��� ∑∞

=

⋅1

!3

nn

n

nn �

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limlim =⋅=⋅=∞→∞→∞→

n

nn

n

n

nn

nnn

nnn

a ��MHU�QH�]QDPR�NROLNR�MH�� n n! . Pokušajmo ovo

riješiti D'Alembertovim kriterijem:�

=+

∞→n

n

n aa 1lim

e

n

nn

nn

nn

nnn

n

n

n

n

n

n

n

3

11

1lim3

)1(3

lim!3

)1()!1(3

lim1

1

=

+

=+⋅=

⋅+

+⋅

∞→∞→

+

+

∞→ �

⇒�� 13 >=e

A ��SD�MH�RYDM�UHG�GLYHUJHQWDQ�����

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