Ejercicios Reglas Derivacion

 

70>

 

Fmrg Narrdra Narrdra - Bapã Bapãtufm tufm 7. Fa gdrivaga gdrivaga

R DBBIÒK 7.  7 . > Dodrbibim 7.9.  Dk baga basm, usd fa gdfikibiòk gd gdrivaga para vdrifibar fas afircabimkds. (i) Fa gdrivaga gd   l ( x) 3  2x  2 x  > ds

g l    3  2. gx

(ii) Fa gdrivaga gd   l ( x) 3  14  14xx + 2> ds

g l    3 14. gx

(iii) Fa gdrivaga gdrivaga gd   l ( x) 3  2x  2 x> ds   l ‾ ( x) 3    4 x   ds   l ‾ ( x) 3 < x  4. (v) Fa gdrivaga gd   l ( x) 3  x 2 +  ds   l ‾ ( x ) 3 2x  2 x> + 1>  1>xx.

Dodrbibim 7.10.  Dk baga basm, bafbufd fa gdrivaga dk df puktm ikgibagm. (i)   l ( x) 3 7x  7 x> ,   l ‾ (>) (ii)   s (t) 3 >.2 >.2tt> ,   s‾ (1.4) (iii)   h(t) 3  7t  7 t>  2,

g h gt

 

t 37

  gc (iv)   c( p) 3 7 p  7 p +  p > , g p

 

 p3>

  gs (v)   s (t) 3  t  007 y >005 pudgd pudgd sdr cmgdfaga pmr  p(t) 3 0.009 0.009tt> + 0.1>  0.1>tt + 1.19 cifds gd hafm hafmkds kds gmkgd   t  0 ds df kÿcdrm kÿcdrm gd añms gdsgd gdsgd df >007. (i) Bafbufd Bafbufd fa baktigag baktigag bmksuciga bmksuciga gd bmcnusti bmcnustinfd nfd df >008. (ii) Gdtdrcikd Gdtdrcikd fa lòrcufa para para fa gdrivaga gd   p. (iii) ½Buáf ds fa razòk gd bacnim gd fa baktigag bmksuciga gd bmcnustinfd df añm >008=

 

Fmrg Narrdra - Rdbbiòk 7.8. Dodrbibims prmpudstms y sus rdspudstas

 

702

Dodrbibim 7.1>. (Baãga gd uk mnodtm).   Rd gdoa badr uk mnodtm gd uk dgifibim dgifibim.. Ihkmrakgm fa rdsistdkbia gdf aird, fa aftura gdf mnodtm   t  sdhukgms gdspués gd gdoarsd badr, dstá cmgdfaga pmr 1 + 100 cdtrm cdtrmss e(t) 3 1< (i) Tsd fa gdfikibiòk gd gdrivaga para eaffar fa ddbuabiòk buabiòk gd fa razòk gd bacnim. (ii)) Tsd fa rdsp (ii rdspuds udsta ta gd fa partd partd (i) para para bafbufa bafbufarr fa rapigdz rapigdz gd uk uk mnodtm mnodtm 1 sdhukgm gdspués gd gdoarsd badr.

Dodrbibim 7.12. (Prdbim gd uka bacisa dk Hacarra).  Df prdbim prmcdgim gd uka bacisa dk Hacar cisa Hacarra ra dktr dktrdd fms añms añms >000 >000 y >004 pudgd pudgd sdr cmgdfagm cmgdfagm pmr pmr    z    b    m . 

>

tt 

   d     p      k    m  c

gmkgd   t     0 ds df kÿcdrm kÿcdrm gd gd añms gdsgd gdsgd   ikibimss gdf ikibim gdf >000.

(i (i)) Tsd Tsd fa gd gdfik fikib ibiò iòk k gd gd gdri riva vaga ga para para ea eaff ffar ar uka uka lò lòrc rcuf ufaa gd fa razò razòk k gd bacn bacnim im gdf prdbim gd uka bacisa. (ii) ½Buáf lud lud fa razòk gd bacnim bacnim gdf prdbi prdbim m gd uka bacisa bacisa a ikibims gdf gdf >002=

Dodrbibim 7.17.  Tk bdktrm bmcdrbiaf tidkd uka bmkburrdkbia bmkburrdkbia gd 1>0 000 visitaktds dk df cds gd su ikauhurabiòk, fudhm fa bmkburrdkbia gd pÿnfibm sd cmgdfa cdgiaktd  >000tt> pdrsmkas  p(t) 3  1>0 000 + >000 gmkgdd 0     t     10 dstá gmkg dstá dk cdsd cdsdss y   t   3  0 bmrrdspmkgd af cds gd fa ikauhurabiòk.

 

     d         g .          e      h         r      d         n          k      m  v 

(i) Bafbufd Bafbufd fa baktigag baktigag gd visitaktd visitaktdss gdspués gdspués gd 10 cdsds. (ii) Bafbufd Bafbufd fa razòk gd bacnim bacnim

g p  . gt

(iii) Bafbufd Bafbufd fa razòk gd bacnim bacnim dk   t  3  9. (iv) Dxpfiqud Dxpfiqud df sihkifibagm sihkifibagm gd gd su rdspudsta rdspudsta gdf gdf ãtdc (iii).

 

707

 

Fmrg Narrdra Narrdra - Bapã Bapãtufm tufm 7. Fa gdrivaga gdrivaga

Dodrbibim 7.14. (Prdbim gd uk nmfdtm).   Df prdbim prmcdgim, prmcdgim, dk kudvms smfds, gd uk  nmfdtm para uk dvdktm gdpmrtivm,   x   añms gdspués gdspués gd 1990 dstá cmgdfagm cmgdfagm pmr  p( x ) 3  9.71  0.19 0.19xx + 0.09  0.09xx> (i) Bafbufd fa razòk gd bacnim gdf prdbim prdbim prmcdgim,

g p  . gx

(ii) ½Buáf lud lud df prdbim prdbim prmcdgim prmcdgim gdf gdf nmfdtm nmfdtm dk df >010= (iii) ½Buáf lud fa razòk razòk gd bacnim gdf prdbim prdbim prmcdgim prmcdgim gd baga baga nmfdtm df >010=

Dodrbibim 7.1)

 y  l    (x    3

(

(i) Hrafiqud fa lukbiòk lukbiòk gdrivaga. gdrivaga. (-7,  0 (

(ii) ½Para qué vafmrds gd   x   dktrd   x   3 7 y   x   3  < fa lukbi lukbiòk òk km ds gdri gdri-vanfd=

0

 x

1 (1,-  > (

()

(7,-  > (

Dodrbibim 7.18. (Msbifabiòk).  A bmktikuabiòk akafizardcms df basm gd uka lukbiòk qud km ds gdrivanfd. Dstm sd eabd cdgiaktd msbifabimkds. Rda  l ( x) 3

 

 1   si   x̵3 0 x 0 si   x  3  0

x sdk

(i) Cudstrd qud   l l    ds bmktikua dk   x  3  0.  l (0 + e)    l (0)  1   3  sdk  . e e  l (0 + e)    l (0) (iii) Dxpfiqud ½pmr qué df fãcitd f·ıc ıc   km dxistd= e 0 e (ii) Cudstrd Cudstrd qud

(iv) Gdbiga si   l l    agcitd gdrivaga fatdraf dk   x  3  0. (v) A bmktikuabiòk bmksigdrd bmksigdrd fa lukbiòk  h( x) 3

 

 1   si   x̵3  0 x 0 si   x  3  0

x> sdk

Tsd fa gdfikibiòk gd fa gdrivaga para cmstrar qud   h  ds gdrivanfd dk   x  3  0 y qud   h‾ (0) 3  0.

 

Fmrg Narrdra - Rdbbiòk 7.8. Dodrbibims prmpudstms y sus rdspudstas

 

704

R DBBIÒK 7.  7 . 2 Dodrbibim 7.15.  Dk baga basm, utifibd fa gdfikibiòk para bafbufar   l ‾ ( x). (ii)   l ( x) 3  x > + 4

(i)   l ( x) 3  x  7

(iii)   l ( x) 3  x 2 + (t> + >)> cdtrms/s gmkgd   t  ds df tidcpm dk sdhukgms. Wmcd fa gd gdri riva vaga ga gd dsta dsta vdfm vdfmbi biga gag g para para ba bafb fbuf ufar ar fa abdfdrabiòk dk df ikstaktd   t  3  > s.  

akzdvd.bmc

 Rupmkha kha qud Dodrbibim Dodrb ibim 7.>1. (Bmstm (Bmstm tmtaf tmtaf). ).  Rupm uka dcprdsa gdtdrcika qud df bmstm dk gòfards gd prmgubir   x  tdfélmkms bdfufards ds gagm pmr B ( x) 3 0.04 0.04xx> + 40  40xx B (201)  B (200)   d iktdrprdtd ds201  200 td rdsuftagm.

Bafbufd

 

     d         g   .       s       d       k      m         e      p            a        t       a         g         i      k u 

Dodrbibim 7.>>.  Dk baga basm, utifibd fa gdfikibiòk para bafbufar   l ‾ ( x). (i)   l ( x) 3  8 (1.2) x + d x

(ii)   l ( x) 3  7 fk x + dπ 

(iii)   l ( x) 3 >sdk x

vafmr luturm luturm gd 1000 kudvms smfds,  smfds,   t  añms gdsDodrbibim 7.>2. (\afmr (\afmr luturm).   Df vafmr pués gd ikvdrt ikvdrtigms igms af af 8 % gd iktdrés iktdrés bmktikum bmktikum ds  1000dd0.08t kudvms smfds  l (t) 3  1000 (i) Dsbrina Dsbrina fa razòk gd bacnim para fa lukbiòk vafmr luturm. luturm. (ii) Bafbufd Bafbufd fa razòk gd bacnim bacnim gdf vafmr vafmr luturm luturm gdspué gdspuéss gd 10 añms.

 

70<

 

Fmrg Narrdra Narrdra - Bapã Bapãtufm tufm 7. Fa gdrivaga gdrivaga

Dodrbibim 7.>7. (Pdsm gd uk ratòk).  Dk uk fanmratmrim sd dstica qud df pdsm gd uk ratòk dktrd dktrd 2 y 11 sdcakas gd dgag pudgd pudgd sdr sdr cmgdfag cmgdfagm m pmr pmr  8.28 fk t  hracms w(t) 3 91.2 + 8.28 gmkgd fa dgag gdf ratòk ds gd   t + > sdcakas sdcakas.. (i) ½Buáf ½Buáf ds df pdsm pdsm gd uk ratòk gd 9 sdcakas sdcakas gd dgag= dgag= y ½qué tak tak rápigm rápigm bacnia su pdsm= (ii) ½Buáf ds fa razòk gd bacnim prmcdgim dk df pdsm gd uk ratòk dktrd fas sdcakas 8 y 11=

Dodrbi Dod rbibim bim 7.>4. 7.>4. (A (Aucd ucdktm ktm gd fa casa). casa).   Rd acasa acasa earik earikaa bm bmk k fd fdva vagu gura ra pmr pmr uk ti tidc dc-pm gd gms emras. emras. Fudhm gd aprmxi aprmxicagacd cagacdktd ktd 7> cikutm cikutmss (gd eandr tdrcikag tdrcikagm m gd acasar), dsta gupfiba su vmfucdk. Df aucdktm gd vmfucdk pudgd sdr cmgdfagm pmr fa lukbiòk e

v(e) 3  d hracms gmkgd   e   ds df kÿcdrm gd emras fudhm qud fa casa bmcdkzò a aucdktar. (i) ½Buáktms cikutms eay qud gdoar fa casa para qud bmksiha uk vmfucdk gd >.4 hracms hracms== (ii) Dsbrina Dsbrina uka lòrcufa para fa razòk gd bacnim gdf aucdktm aucdktm gd casa.

Dodrbibim 7.>π 

∟  ( ) >π g κ

pid/s

   c    m     b      t .      m     p    s       h      m       f      n .      a     k      i     t     k    d     h    r      a       e      i      g 

gmkgd   h   ds fa fa hravdgag hravdgag (2> pid/s> ) y the ds fa takh takhdkt dktdd eipd eipdrnò rnòfib fiba. a. (i) Tka mfa gd tsukaci gd tsukaci pudgd  pudgd tdkdr   κ  3  2>5 052 pids. Df mbéakm mbéakm tidkd tidkd uka uka prmprmlukgigag lukgi gag prmcdgim prmcdgim gd 1> >00 pids. ½Qué tak rápigm rápigm viaoa uka mfa a través gdf mbéakm= (ii) Wdkidkgm dk budkta qud   κ   ds uk kÿcdrm fiom partibufar, gdtdrcikd gdtdrcikd uka lòrcufa para   v‾ (g). (iii) ½Buáfds ½Buáfds smk fas ukigagds ukigagds para   v‾ (g)  y qué sihkifiba lãsibacdktd=

 

Fmrg Narrdra - Rdbbiòk 7.8. Dodrbibims prmpudstms y sus rdspudstas

 

708

Dodr Dodrbi bibi bimm 7.>8. 7.>8. (Fdy (Fdy gd Pm Pmus usif iffd fd). ).   Gd abudrgm a fa fdy gd Pmusiffd, fa vdfmbigag (dk bdktãcdtrms pmr sdhukgm) gd fa sakhrd a   r bc gdf dod bdktraf gd uka artdria ds gaga pmr v(r ) 3  j ( ^>  r> )   0  r  ^ gmkgd   j  ds j  ds uka bmkstaktd y   ^  ds df ragim   apmtedjdk-ucsbeau.gd gd fa artdria. Cudstrd qud df ﬎uom sakhuãkdm ds cás rápigm dk df dod bdktraf. ½Gòkgd df ﬎uom sakhuãkdm ds cás fdktm=

Dodrbibim 7.>5. (^dabbiòk gdf budrpm a fa cdgibika).   Fa rdabbiòk gdf budrpm a uka gmsis gd cdgibika pudgd sdr dxprdsaga cdgiaktd fa lòrcufa B

^  3  C >

Z

>

 

 C 2

S

gmkgd   B  ds uka baktigag pmsitiva y   C  ds fa baktigag gd cdgibika ansmrniga dk fa sakhrd. Ri fa rdabbiòk ds uk bacnim gd prdsiòk sakhuãkda,   ^   sd cigd dk cifãcdtrms gd cdrburim. Ri fa rdabbiòk ds uk bacnim gd tdcpdratura,   ^   sd cigd dk hragms hragms.. Bafbufd Bafbufd g^/g /g C gdriva ivaga ga bmcm lukbi lukbiòk òk gd   C. Ds Dsta ta ds  C , qud ds fa gdr ffacaga fa sdksinifigag gd fa sakhrd a fa cdgibika.

Dodrbibim 7.>9. (Prdsiòk dk uk bifikgrm).  Ri uk has dk uk bifikgrm sd caktidkd a uka tdcpdratura bmkstaktd   W , fa prd prdsiò siòk k   P  sd rdfabimka bmk df vmfucdk   \  cdgiaktd fa lòrcufa  ak  a k>   k^W    > P  3 \   kn \  dk gmkgd   a,   k,   n   y   ^   smk bmkstakt bmkstaktds. ds. Bafbufd Bafbufd g P/g /g\  \ .

Dodrbibim 7.20. (Ikhrdsm carhikaf).  Rupmkha qud df ikhrdsm sdcakaf dk gòfards gd fa vdkta gd   x   dsbritmrims gd mfibikas, edbems a cdgiga ddss

(

  1 I ( x) 3  >000 1  x + 1

)

(i) Eaha fa hráfiba gd   I . ½Para qué qué vafmrds vafmrds gd   x  tidkd sdktigm df prmnfdca= (ii) Bafbufd df ikhrdsm ikhrdsm carhikaf buakgm sd vdkgdk   x   dsbritmrims. (iii) Tsd fa lukbiòk   I ‾ ( x)  para dsticar df ikbrdcdktm dk df ikhrdsm buakgm fas vdktas aucdktak aucdktak gd gd 4 dsbritmrims dsbritmrims sdcakafds sdcakafds a < dsbrit dsbritmrims mrims sdcakafds. sdcakafds.

 

705

 

Fmrg Narrdra Narrdra - Bapã Bapãtufm tufm 7. Fa gdrivaga gdrivaga

Dodrbibim 7.21. (Bmcdrbiaf gd tdfdvisiòk).  Df bmstm, dk cifds gd kudvms smfds, pmr trakscitir bmcdrbiafds gd W\ gd uka carba gd bdfufar, dsta gagm pmr B ( x) 3 140  + >>40  >>40xx  0.0> 0.0>xx> (i) Gdtdrcikd Gdtdrcikd fa lukbiòk gd bmstm carhikaf carhikaf y ÿsdfa para dsticar dsticar bmk qué rapigdz aucdkta df bmstm buakgm   x   3  7. Bmcpard Bmcpard dstm bmk bmk df bmstm dxabtm dxabtm gd trakscitir df quiktm bmcdrbiaf. (ii) Bafbufd fa lukbiòk lukbiòk bmstm prmcdgim prmcdgim   B  y dvafÿd   B (7). ½Qué ffdd gibd df df rdsufrdsuftagm=

Dodrbibim 7.2>. (Prmgubbiòk g gdd pdfubeds).  Df bmstm gd prmgubir   x  msitms gd pdfubed af gãa, dk uka dcprdsa, lud bafbufagm pmr df gdpartacdktm gd cdrbagdm y sd mntuvm fa lòrcufa  70xx  0.001 0.001xx> B ( x) 3  100  + 70 (i) Gdtdrcikd Gdtdrcikd fa lukbiòk gd bmstm carhikaf carhikaf y ÿsdfa para dsticar dsticar bmk qué rapigdz aucdkta df df bmstm a uk kivdf gd prmgubbiòk prmgubbiòk gd 100 msitms. Bmcpard dstm bmk df bmstm dxabtm dxabtm gd prmgubir prmgubir df msitm msitm 101. (ii) Gdtdrcikd Gdtdrcikd fa lukbiòk bmstm prmcdgim prmcdgim   B  y dvafÿd   B (100). ½Qu ½Quéé fd gibd gibd df rdsuftagm=

Dodrbibim 7.22. (Ikhrdsm carhikaf? prdbim gdf atÿk).  Rupmkha qud fa dbuabiòk gd fa gdcakga gdf atÿk dk uk pdqudñm pudnfm bmstdrm dstá gaga pmr  p  3

 >0 000   q1.4

bmk

>00  q  500

gmkgd   p   ds df prdbim, dk kudvms smfds, pmr pmr jifmhracm gd atÿk, y   q  ds fa baktigag gd jifmhracms gd atÿk qud sd pudgdk vdkgdr af prdbim   p  dk uk cds. (i) Bafbufd df prdbim qud fa ikgustria pdsqudra gdnd bmnrar pmr jifmhracm gd atÿk para tdkdr tdkdr uka gdcakga gdcakga gd 700 jifmhracms jifmhracms cdksuafds cdksuafds gd atÿk. atÿk. (ii) Bafbufd Bafbufd df ikhrdsm cdksuaf cdksuaf   I   dk lukbiòk gd   q, fa bakti baktigag gag gd jifmhra jifmhracms cms I  dk gd atÿk. (iii) Bafbufd Bafbufd df ikhrdsm ikhrdsm y df ikhrdsm ikhrdsm carhikaf carhikaf buakgm buakgm sd gdcakgak gdcakgak 700 jifmhr jifmhraacms cdksuafds d iktdrprdtd fms rdsuftagms. (iv)) Ri fa baptura (iv baptura cdksuaf cdksuaf dk fa ikgustri ikgustriaa pdsqu pdsqudra dra ds gd 700 jif jifmhr mhracm acmss gd atÿk, y df prdbim prdbim dstá dk df vafmr bafbufagm bafbufagm gd fa partd (i), ½rdbmcdkg ½rdbmcdkgarãa arãa ustdg a fa ikgustria pdsqudra qud aucdktd m naod su prdbim para aucdktar su ikhrdsm=