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&iOFXOR�, Autoria: Jeanne Dobgenski Tema 08 Conceito de Derivada 7HPD��� Conceito de derivada Autoria: Jeanne Dobgenski Como citar esse documento: DOBGENSKI, Jeanne. Cálculo I: Conceito de Derivada. Caderno de Atividades. Valinhos: Anhanguera Educacional, 2014. Índice ������$QKDQJXHUD�(GXFDFLRQDO�� 3URLELGD� D� UHSURGXomR� ¿QDO� RX� SDUFLDO� SRU� TXDOTXHU�PHLR� GH� LPSUHVVmR�� HP� IRUPD� LGrQWLFD�� UHVXPLGD� RX�PRGL¿FDGD� HP� OtQJXD� SRUWXJXHVD�RX�TXDOTXHU�RXWUR�LGLRPD� Pág. 16 Pág. 17 Pág. 17 Pág. 16 Pág. 14Pág. 13 ACOMPANHENAWEB Pág. 3 CONVITEÀLEITURA Pág. 3 PORDENTRODOTEMA � Conceito de Derivada No tema anterior, você estudou que se o limite de função f(x) existe, então a função tende a um valor “L” quando x tende a um valor “c”, .)(lim Lxf cx o (VVH�FiOFXOR�SRGH�VHU�HIHWXDGR�DR�YHUL¿FDU�R�YDORU�GD�WD[D�GH�YDULDomR�PpGLD�HP�LQWHUYDORV�FDGD�YH]�PHQRUHV�GH�IRUPD� a x�VHU�VX¿FLHQWHPHQWH�SUy[LPR�GH�F��(VVH�SURFHVVR��RX�WLSR�GH�OLPLWH��IRL�LGHQWL¿FDGR�FRPR�R�FiOFXOR�GD�WD[D�GH�YDULDomR� LQVWDQWkQHD�RX��DLQGD��FRPR�D�GHWHUPLQDomR�GR�FRH¿FLHQWH�DQJXODU��LQFOLQDomR��GD�UHWD�WDQJHQWH�TXH�SDVVD�QR�SRQWR� OLPLWH�H��DLQGD�PDLV��HVVD�p�D�GH¿QLomR�GD�GHULYDGD�QXP�SRQWR� 6HJXQGR�+XJKHV�+DOOHWW�et al���������S������� ������D� WD[D�GH�YDULDomR�PpGLD�QRV�GL]�R�TXmR�GHSUHVVD��RX�GHYDJDU��D� IXQomR�PXGD��GH�XPD�H[WUHPLGDGH�GR� LQWHUYDOR�DWp�D�RXWUD��HP�UHODomR�DR�WDPDQKR�GR�LQWHUYDOR��e�PDLV�~WLO��PXLWDV�YH]HV��VDEHU�D�WD[D�GH�YDULDomR�GR� TXH�D�YDULDomR�DEVROXWD��3RU�H[HPSOR��VH�DOJXpP�OKH�RIHUHFH�XP�HPSUHJR�TXH�SDJD�������YRFr�YDL�TXHUHU�VDEHU� TXDQWR�WHPSR�YDL�WHU�TXH�WUDEDOKDU�SDUD�JDQKDU�HVVH�GLQKHLUR��1mR�EDVWD�VDEHU�DSHQDV�D�YDULDomR�WRWDO�HP�VHX� GLQKHLUR��������PDV�VH�VRXEHU�D�WD[D�GH�YDULDomR��LVWR�p�������GLYLGLGR�SHOR�WHPSR�TXH�YDL�OHYDU�SDUD�UHFHEr�OR�� YRFr�SRGH�GHFLGLU�VH�DFHLWD�RX�QmR�R�HPSUHJR�� 2EVHUYH�TXH�HVVH�WH[WR�PRVWUD�XPD�DSOLFDomR�GH�WD[D�GH�YDULDomR�PpGLD�TXH�YRFr�Mi�VDEH�TXH�JHRPHWULFDPHQWH�p�XPD� UHWD�VHFDQWH�j�IXQomR��1HVWH�WHPD��YRFr�YHUi�TXH�D�WD[D�GH�YDULDomR�LQVWDQWkQHD��Mi�FRPSUHHQGLGD�FRPR�R�SURFHVVR�GH� WRPDU�XP�OLPLWH��WDPEpP�SRGH�VHU�FRQVLGHUDGD�FRPR�D�GHULYDGD�QXP�SRQWR��2EVHUYH�TXH�VHXV�FRQKHFLPHQWRV�HVWmR� sendo ampliados! Aproveite a leitura! CONVITEÀLEITURA C it d D i d sendo ampliados! Aproveite a leitura! 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TXH�p�QHFHVViULR�VXEWUDLU�D�IXQomR�f(x) quando estiver no ponto x� �D���K�GD�f(x) quando x� �D��/RJR��DOJHEULFDPHQWH�D� VROXomR�p�D�GHVFULWD�D�VHJXLU� 82)(' )82(lim 82 lim)(' 989882 lim)(' ]98[]9)(8)[( lim)(' )()( lim)(' 0 2 0 222 0 22 0 0 � �� �� ������� ������� �� oo o o o aaf ha h hhahaf h aahahahaaf h aahahaaf h afhafaf hh h h h PORDENTRODOTEMA � Será que para calcular a derivada de uma função será sempre necessário calcular o limite da função naquele ponto? 1mR��LVVR�SRGHUi�VHU�IHLWR�SRU�PHLR�GH�UHJUDV�Mi�GHWHUPLQDGDV�H�TXH�DX[LOLDUmR�QR�FiOFXOR�GDV�GHULYDGDV��3ULPHLUDPHQWH�� QR�HQWDQWR��p�LPSRUWDQWH�H[SORUDU�D�GHULYDGD�FRPR�D�LQFOLQDomR�GD�UHWD�WDQJHQWH�H�FRPSUHHQGHU�TXH�D�GHULYDGD�WDPEpP� SRGH�VHU�YLVWD�FRPR�XPD�IXQomR� Exemplo 2�±�(QFRQWUDU�XPD�HTXDomR�GD�UHWD�WDQJHQWH�j�SDUiEROD�\� �[��±��[�����QR�SRQWR���������67(:$57��������S������� Solução��OHPEUH�VH TXH�D�HTXDomR�GD�UHWD�p�GDGD�SRU�\� �m[���E��VHQGR�m a inclinação da reta e b o ponto que essa reta corta o eixo y�� &RQVLGHUH�R�SRQWR��D��f�D����RX�VHMD��D�coordenada (x, y��p�UHSUHVHQWDGD�SRU�x� �D�H�y� �I�D�� &RPR�IRL�YLVWR�D�LQFOLQDomR�GD�UHWD�WDQJHQWH�QXP�SRQWR�GD�FXUYD�p�D�GHULYDGD�GD�IXQomR�QHVVH�SRQWR��HQWmR�P� �I¶�D���1R� H[HPSOR���IRL�HQFRQWUDGD�D�GHULYDGD�GD�IXQomR�f(x�� �x��±��x�����QR�SRQWR�³D´�FRPR�f¶�D�� ��D�±���� /RJR��D�LQFOLQDomR�GD�UHWD�WDQJHQWH�QR�SRQWR���������p�HQFRQWUDGD�DR�VXEVWLWXLU�R�YDORU�GHVVH�SRQWR�QD�IXQomR�GHULYDGD�� f¶���� ��������� �����'HVVD�IRUPD��P� �f¶���� ����H�D�HTXDomR�GD�UHWD�SRGH�VHU�HVFULWD�FRPR�y� ���x���E� 0DV�FRPR�FDOFXODU�R�YDORU�GH�E�TXH�p�R�SRQWR�QR�HL[R�y�SHOR�TXDO�D�UHWD�WDQJHQWH�SDVVD"�6H�D�UHWD�WDQJHQWH�SDVVD�QR� SRQWR���������HQWmR�VXEVWLWXD�HVVHV�SRQWRV�QD�HTXDomR�HQFRQWUDGD� y� ���x���E ��� ����������E E� ��������ĺ�E� �� 3RUWDQWR�D�HTXDomR�GD�UHWD�WDQJHQWH�TXH�SDVVD�QR�SRQWR���������GD�IXQomR�f(x�� �x��±��x�����p�H[SUHVVD�SRU�y� ���x� 2XWUD�IRUPD�GH�HQFRQWUDU�D�HTXDomR�GD�UHWD�WDQJHQWH�j�FXUYD�y� �f(x) no ponto (a, f�D���p�FDOFXODU�y�±�f�D�� �f¶�D��x ��D��� 2X�VHMD��R�FiOFXOR�VHULD� y�±�I�D�� �I¶�D��x�±�D� y�±������ ����[�±��� y����� ���x���� y� ���x������ĺ�\� ���x� PORDENTRODOTEMA � $¿UPD�VH�TXH�XPD� IXQomR� I�p�GHULYiYHO� �RX�GLIHUHQFLiYHO�� VH��SUy[LPR�GH�FDGD�SRQWR� ³D´�GR�VHX�GRPtQLR��D� IXQomR� I�[��í�I�D��VH�FRPSRUWDU�DSUR[LPDGDPHQWH�FRPR�XPD�IXQomR�OLQHDU��RX�VHMD��VH�R�VHX�JUi¿FR�IRU�DSUR[LPDGDPHQWH�XPD� UHWD��2�GHFOLYH�GHVVD�UHWD�p�D�GHULYDGD�GD�IXQomR�f no ponto a.�(VVD�UHWD��WDQJHQWH��QDV�SUR[LPLGDGHV�GH�D��³VH�FRQIXQGH� FRP�D�FXUYD´��SRGHQGR�³GH�FHUWD�IRUPD´�VXEVWLWXt�OD��)LJXUD������ )LJXUD�������$PSOLDomR�GD�UHWD�WDQJHQWH�DR�SRQWR�3� )RQWH��0XUROR�H�%RQHWWR��������S������� Derivada como função $Wp�DJRUD�D�GHULYDGD�GH�XPD�IXQomR�IRL�HVWXGDGD�HP�XP�SRQWR�¿[R��&RQVLGHUH��DJRUD��R�TXH�DFRQWHFH�HP�XPD� série�GH�SRQWRV��$�GHULYDGD��HP�JHUDO��DVVXPH�YDORUHV�GLIHUHQWHV�HP�SRQWRV�GLIHUHQWHV�H�p��WDPEpP��XPD�IXQomR��(P� SULPHLUR�OXJDU��OHPEUH�VH�GH�TXH�D�GHULYDGD�GH�XPD�IXQomR�HP�XP�SRQWR�PRVWUD�D�WD[D�VHJXQGR�D�TXDO�R�YDORU�GD�IXQomR� HVWi�YDULDQGR�QDTXHOH�SRQWR��*HRPHWULFDPHQWH��D�GHULYDGD�SRGH�VHU�FRQVLGHUDGD�D�LQFOLQDomR�GD�FXUYD�RX�R�FRH¿FLHQWH� DQJXODU�GD�UHWD�WDQJHQWH�j�FXUYD�QR�SRQWR��FRQIRUPH�H[SOLFD�+XJKHV�+DOOHWW�et al.��������S������ Exemplo 3� ��Estimar a derivada da função f(x), FXMR�JUi¿FR�DSDUHFH�QD�)LJXUD������SDUD�x ����� ��������������������� �+8*+(6�+$//(77�et al.��������S������ PORDENTRODOTEMA � )LJXUD�����±�'HULYDGD�YLVWD�JUD¿FDPHQWH�FRPR�R�FRH¿FLHQWH�DQJXODU�GD�UHWD�WDQJHQWH� )RQWH��+XJKHV�+DOOHWW�et al. �������S������ Solução��D�SDUWLU�GR�JUi¿FR��p�SRVVtYHO�HVWLPDU�D�GHULYDGD�HP�TXDOTXHU�SRQWR�WUDoDQGR�D�UHWD�WDQJHQWH�QDTXHOH�SRQWR�H� HVWLPDQGR�R�FRH¿FLHQWH�DQJXODU�GD�WDQJHQWH��SRU�PHLR�GR�XVR�GH�SDSHO�TXDGULFXODGR��FRPR�QR�H[HPSOR�GD�)LJXUD������� 3RU�H[HPSOR��D�UHWD�WDQJHQWH�HP�x ����WHP�FRH¿FLHQWH�DQJXODU�SHUWR�GH����GH�PRGR�TXH�f’�����§����1RWH�TXH�D�LQFOLQDomR� em x �����p�SRVLWLYD�H�EHP�JUDQGH��D�LQFOLQDomR�HP�x ����p�SRVLWLYD��PDV�PHQRU��(P�[� ����D�LQFOLQDomR�p�QHJDWLYD�H�� HP�[� ����PDLV�QHJDWLYD�DLQGD��(VVD�DQiOLVH�SRGH�VHU�IHLWD�SDUD�WRGRV�RV�SRQWRV��/RJR��REVHUYH�TXH�SDUD�WRGR�YDORU�GH� x�H[LVWH�XP�YDORU�FRUUHVSRQGHQWH�SDUD�D�GHULYDGD��2X�VHMD��D�GHULYDGD�p�XPD�IXQomR�GH�[�� $�)LJXUD�����DSUHVHQWD�YDORUHV�HVWLPDGRV�SDUD�D�GHULYDGD�QRV�SRQWRV�LQGLFDGRV�QR�HQXQFLDGR��7UDoDQGR�DV�WDQJHQWHV� DRV�SRQWRV�QR�JUi¿FR��YHUL¿FD�VH�TXH�RV�YDORUHV�HQFRQWUDGRV�VmR�VHPHOKDQWHV�DRV�PRVWUDGRV�)LJXUD�����±�9DORUHV�HVWLPDGRV�SDUD�D�GHULYDGD�GD�IXQomR� )RQWH��+XJKHV�+DOOHWW�et al. �������S����� PORDENTRODOTEMA � +i�PXLWDV�QRWDo}HV�XVDGDV�SDUD�UHSUHVHQWDU�D�GHULYDGD�GH�XPD�IXQomR�y �f(x). $OpP�GH�f’(x), DV�PDLV�FRPXQV�VmR� ).()()(')(' xfDxDfxf dx d dx df dx dyyxf x Os operadores D e d/dx VmR�FKDPDGRV�RSHUDGRUHV�GLIHUHQFLDLV��SRLV� LQGLFDP�D�RSHUDomR�GH�GLIHUHQFLDomR�TXH�p�R� SURFHVVR�GH�FiOFXOR�GH�XPD�GHULYDGD��dy/dx�p�OLGR�FRPR�³D�GHULYDGD�GH�y em relação a x”, e df/dx ou (d/dx)f(x) como “a derivada de f em relação a x´� As notações�TXH� LQGLFDP�D�GHULYDGD�GH�XPD� IXQomR� WDPEpP�SRGHP� LQGLFDU�XP�SRQWR�HP�TXH�VH�GHVHMD�DYDOLDU�D� GHULYDGD��FRPR�VHJXH� .)(' axax ax xfdx dou dx dyouy 2�VtPEROR�GH�DYDOLDomR��_ [ D ��VLJQL¿FD�FDOFXODU�D�H[SUHVVmR�j�HVTXHUGD�HP�x� �a�� $JRUD�TXH�YRFr�Mi�FRQKHFH�DV�QRWDo}HV�SDUD�DV�GHULYDGDV�GH�IXQo}HV��DSUHQGHUi�DOJXPDV�UHJUDV�GH�GHULYDomR��(VVDV� UHJUDV�SHUPLWHP�FDOFXODU�D�GHULYDGD�GH�XPD�IXQomR�UDSLGDPHQWH��VHP�SUHFLVDU�XVDU�D�GH¿QLomR�GH�GHULYDGD�FRPR�IRL� IHLWR�QR�([HPSOR��� Regra 1 – derivada de uma função constante é zero. .0)( c dx d Exemplo 4�±�FDOFXOH�D�GHULYDGD�GH�f(x�� ��� Solução:�REVHUYH�TXH�HVVD�p�XPD�IXQomR�FRQVWDQWH�TXH�SDVVD�QR�SRQWR���GR�HL[R�\�H�QmR�FRUWD�R�HL[R�x��PDV�p�SDUDOHOR� D�HOH��/RJR�HVVD�UHWD�p�KRUL]RQWDO�H�QmR�WHP�LQFOLQDomR��FRH¿FLHQWH�DQJXODU� �P� �����,VVR�VLJQL¿FD�TXH�DR�YDULDU�R�YDORU� em x�QmR�Ki�DOWHUDomR�HP�y��ORJR�QmR�H[LVWH�XPD�WD[D�GH�YDULDomR�LQVWDQWkQHD�SDUD�DOJR�FRQVWDQWH��&RQVHTXHQWHPHQWH�� D�GHULYDGD�GH�XPD�IXQomR�FRQVWDQWH�p�]HUR� PORDENTRODOTEMA � Regra 2 – derivada de uma função potência, quando n for um número real qualquer. .)( 1� nn nxx dx d Exemplo 5 – calcule a derivada de f(x�� �x���(P�VHJXLGD��GHWHUPLQH�D�WD[D�GH�YDULDomR�LQVWDQWkQHD�GD�IXQomR�f(x) em x� ��� Solução: Regra 3 – derivada de uma função multiplicada por constante. ).()]([ xf dx dcxcf dx d Exemplo 6 – calcule a derivada de f�[�� ���x���(P�VHJXLGD��GHWHUPLQH�D�WD[D�GH�YDULDomR�LQVWDQWkQHD�GD�IXQomR�I�[��HP�x� ��� Solução: .480)4('16.304.30)4(' 30)('3.10)('10)('10)(' 2 21333 � ff xxfxxfx dx dxfx dx dxf Regra 4 – derivada da soma ou diferença de duas funções deriváveis. ).()()]()([ xg dx dxf dx dxgxf dx d r r Exemplo 7 – calcule a derivada de f�[�� �x��±��x������(P�VHJXLGD��GHWHUPLQH�D�WD[D�GH�YDULDomR�LQVWDQWkQHD�GD�IXQomR� f(x) em x� ����PHVPD�VLWXDomR�GR�H[HPSOR���� PORDENTRODOTEMA f x x x x x x xf f f f �� Solução: .283.2)3(' 82)(' 082)(' 98)(' 98)(' )98()(' 1112 2 2 2 � � � �� �� �� �� �� f xxf xxxf dx dx dx dx dx dxf dx dx dx dx dx dxf xx dx dxf ([LVWHP�UHJUDV�SDUD�GHULYDU�GLYHUVRV�WLSRV�GH�IXQomR��FRPR�SRGH�VHU�YLVWR�QDV�)LJXUDV�����H������3RUWDQWR�QmR�GHL[H�GH� SUDWLFDU�H�FRPSOHPHQWH�VHX�HVWXGR�VREUH�GHULYDGDV� PORDENTRODOTEMA �� )LJXUD�����±�5HJUDV�GH�GHULYDomR�±�IyUPXODV�JHUDLV��IXQo}HV�H[SRQHQFLDLV�H�ORJDUtWPLFDV� )RQWH��6WHZDUW�������� PORDENTRODOTEMA �� )LJXUD�����±�5HJUDV�GH�GHULYDomR�±�IXQo}HV�WULJRQRPpWULFDV�H�KLSHUEyOLFDV� )RQWH��6WHZDUW�������� PORDENTRODOTEMA �� Limites � %RP�PDWHULDO�VREUH�R�HVWXGR�GDV�GHULYDGDV� 'LVSRQtYHO�HP��<KWWS���ZZZ�XIUJV�EU�OPTD�DUTXLYRV�XSORDGV�/,0,7(6�H�'(5,9$'$6�SGI>��$FHVVR�HP����MXQ������� Entendendo o que é a derivada � 3iJLQD�TXH�FRQWpP�H[SOLFDo}HV�VREUH�DV�GHULYDGDV��7H[WR�FXUWR��PDV�EDVWDQWH�FODUR� 'LVSRQtYHO�HP��<KWWS���ZZZ�DQGUHPDFKDGR�RUJ�DUWLJRV�����HQWHQGHQGR�R�TXH�H�D�GHULYDGD�KWPO>��$FHVVR�HP�� ��MXQ������� 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UHODomR�D�XP�VLVWHPD�GH�UHIHUrQFLD��FRPR�H[HPSOR��RV�HL[RV�FDUWHVLDQRV��[�H�\�� Notação:�TXDOTXHU�VLVWHPD�GH�VtPERORV�H�DEUHYLDo}HV�TXH�DMXGD�DV�SHVVRDV�D�WUDEDOKDUHP�HP�XP�GHWHUPLQDGR�DVVXQWR�� 2V�PDWHPiWLFRV�XVDP�D�QRWDomR�SDUD�VLPSOL¿FDU�LGHLDV�H�SUREOHPDV� Série:�FRQMXQWR�GH�JUDQGH]DV�RUGHQDGDV�SRU�RUGHP�FUHVFHQWH�RX�GHFUHVFHQWH�Dif iá l f ã f p d i á l ( dif iá l� y L G G W G G t L I m f(f � f(f ) GLOSSÁRIO Série:�FRQMXQWR�GH�JUDQGH]DV�RUGHQDGDV�SRU�RUGHP�FUHVFHQWH�RX�GHFUHVFHQWH� GABARITO Questão 1 Resposta: Taxa de variação média R$/ton 21 5 105 5 3108 5 500-13-500363 V de Taxa 5 500)1(3-50063 16 C(1)C(6) V de Taxa m 22 m � � �� � � /RJR��HP�PpGLD��JDVWD�VH����UHDLV�SRU�WRQHODGD�GH�WULJR�SDUD�ID]HU�R�EHQH¿FLDPHQWR� Inclinação da reta secante:�FRPR�REVHUYDGR�QD�WHRULD��D�LQFOLQDomR�GD�UHWD�VHFDQWH�QR�LQWHUYDOR�GH���DWp���WRQHODGDV�p� R�SUySULR�YDORU�GD�WD[D�GH�YDULDomR�PpGLD��RX�VHMD���P�VHFDQWH� ���� �� Questão 2 Resposta: $OWHUQDWLYD�(� Questão 3 Resposta: $OWHUQDWLYD�'� Questão 4 Resposta: A derivada f(x��p�f '(x�� ���x����x��� 3DUD�x� ����f '���� ��� ������� ������ f ���� ��� ��������� f '(2) = 174. 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