A Matemática Do Ensino Médio Vol 1

November 12, 2022 | Author: Anonymous | Category: N/A

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D @dtf`ätjkd cg Fnsjng @êcjg Xgbu`f 4 Fbgn Bdif Fbgn Bdifss Bj Bj`d `d Qdubg Qdu bg Kfzdr Kfzdr Qjntg Qjntg Kdrvdb Kdrvdbmg mg Fcudrcg Ydinfr Duiust Dui ustg g Kêzd Kêzdrr @gridc @gridcg g

[GBVKJGNÄUJG KG@QBFSG

KGBFÉËG CG QUGHF[ QUGHF[[GU [GU CF @DSF@ @DSF@ÄSJKD ÄSJKD [GKJFCDCF EUD[JBFJUD EUD[JBFJUD CF @DSF@ÄSJKD

[gbu½ kk˒ g ˒fs cg Bj Bjvrg vrg< D @dtf`dtjkd d ³tjkd cg Fnsjng @³ f fcjg cjg - [E@ [E@ (Fbgn Bdifs Bdifs Bj`d f kgb. kgb.)) njeebfcjfigAi`djb.kg` njeebfcjfigA Kg`pjbdcg cjd 5;/1:/5148

Fssf cgku`fntg fst³d f` kgnstdntf rfvjs˒ rfv js˒ddg. g. Xfz gu gu gut gutrd rd u` frrg cf pgrtuiuˈffss ³f kgrrjijcg, u`d pdssdif` quf n˒ ddgg fflkgu ùjtg ùjtg kbdrd ³f rfhfjtd, u`d sgbu½kk˒ ddg ˒g ffqujvgkdc qujvgkdcdd ³f suestjtu³ sue stjtu³ĴĴcd cd gu d sgbu½ sgb u½ k˒ddgg cf u`d cds qufst˒gfs djncd n˒dg n˒dg rfsgbvjcd rfsgbvjcdss dpdrfkf `dijkd`fntf f` `jnm `jnmdd kdef½kd, sfncg jnkbu³ Ĵcd Ĵcd f` vfr vfrs˒ s˒ gfs dtudbjzdcds cg cgku`fntg. Dssj`, vfrjfflquf sf g quf vgkˈf tf` f` `˒dgs ³f cf hdtg d vfrs˒ddgg `djs rfkfntf cg cgku`fntg . Sgcds Sgcds ds dtudbjzd½kk˒gfs ˒gfs cfbf fst˒ ddgg cjspgn³ĴĴvfjs vfjs f` f`   www.nuèfr.=:1`.kg` www.nuèfr.=:1`.kg` sf` er `fs `fs`g. `g.

[f qujsfr jnhgr`dr dbiu` frrg cf pgrtuiuˈffs, s, cjijtd½kk˒ dg d˒g gu `fs`g `fs`g cf b³ gijkd ngs fxfrk³ Ĵkjgs fskrfvd pdrd< njeebfcjfigAi`djb.kg`

[u`³ drjg drjg 4 Kgnluntgs

5

5 N³ u`frgs Ndturdjs

43

; N³ u`frgs Kdrcjndjs 3 N³ u`frgs Ufdjs

55 5=

6 Hun Hun½ ½kg ˒fs Dfflns

;6

8 Hun Hun½ ½kg ˒fs Tudcr³ dtjkds

68

9 Hun Hun½ ½kg ˒fs Qgbjng`jdjs

=1

= Hun Hun½ ½kg ˒fs F Fx xpg nfnkjds f Bgidr³ Ĵt`jkds

=9

: Hun Hun½ ½kg gfs ˒fs Srjig Srjign ng`³ ftrjkds

:6

41 Dirdcfkj`fntg

415

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

4

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

Kgnluntgs

4. [fld` Q 4, Q5, T4 , T5 prgprjfcdcfs rfhfrfntfs d fbf`fntgs fbf`fntgs cf u` kgnluntg unjvfrsg V . [upgnmd quf Q 4 f Q5 fsigtd fsigtd` ` tgcgs tgcgs gs kdsgs pgss³ pgss³ĴĴvfjs vfjs (gu sfld, u` fbf`fntg fbf`fntg qudbqufr qudbqufr cf V   gu tf` prg prgprj prjfcd fcdcf cf Q 4 gu gu tf` ˈf` Q 5 ). [u [upg pgnm nmdd djncd djncd quf quf T 4 f T5 s˒ddgg jjnkg nkg`pd `pdt³ t³ĴĴvfj v fjss (jstg (js tg ³ff,, fxkbuf`-s fxkbu f`-sff ù ùtudb tudb`fntf) `fntf).. [upg [upgnmd, nmd, fflndb`fntf fflndb`fntf,, quf Q 4 T4 f Q5 T5 . Qrgvf Qrgvf quf vdbf` vdbf` ds rfk³ Ĵprg Ĵ prgkd kds< s< T4 Q 4 f T5 Q 5 .

⇕

⇕

⇕

⇕

[gbu½ k˒ dg< kg`gg Q 5 T5, fnt˒dg u` Kg`g Q 4 f Q5 fsigtd` tgcds ds pgssjejbjcdcfs pgssjejbjcdcfs f Q 4 T4 ef` kg` fbf`fntg cf V   gu    tf` ˈf` prgp prgprjfcd rjfcdcf cf T4 gutˈf` f` prgprjfcd prgprjfcdcf cf T5. Gu f` gut gutrds rds pdbdvr pdbdvrds< ds< n˒dg n˒dg pgcf mdvfr fbf`fntg cf V    quf    n˒dg igzf ig zf cf T 4 f T5 dg `fs`g tf`pg.

⇕

⇕

Q5 . Nfstf Nfstf kdsg u u` ` fbf`fntg fbf`fntg u pfrtfnkfntf d V    tˈ f` td td` è³ f` f` [upgnmd pgr desurcg quf T 4 prgprjfcdcf T5 , pg pgjs js Q 5 T5 . G quf ifr ifrdd u` desur desurcg cg l³d quf T4 f T5 sf fxkbuf` ùtudb`fntf. Bgig T 4 Q 4.

⇕

⇕

⇕

Dndbgid`fntf sf prgvd quf T5

⇕ Q5.

5. Fnqudcrf ng kgntfxtg cg fxfrk³ĴĴkjg kjg dntfrjgr g sfiujntf hdtg ifg`³ftrjkg< ftrjkg< Cuds Cuds geb geb³ ³ Ĵqu Ĵquds ds quf sf dhdstd dhdstd` ` jiudb` jiudb`fnt fntff cg p³ f cd pf pfrpfnc rpfncjkubd jkubdrr sd dg ˒g ji jiudj udjs. s. [f sf dhdst dhdstd` d` cfsjiu cfsjiudb`f db`fntffnt˒ ntffnt˒dg s˒dg dg cf cfsj sjiu iudj djss f d `d `djg jgrr ³ f d quf `djs `djs sf dhds dhdstd td.. [gbu½ k˒ dg< Hdzfncg u`d kg`pdrd½kk˒ dg d˒g kg` g fxfrk³Ĵkjg Ĵkjg dntfrjgr dntfrj gr tfrf`gs<

Q4 < Qrgprjfcdcf cf sf dhdstdr dhdstdr jiudb`fntf. T4< Qrgpr Qrgprjfcdc jfcdcff cf sfrf` cf td`dnmgs td`dnmgs jiudjs. Q5 < Qrgprjfcdcf cf sf dhdstdr cfsjiudb`fntf. cfsjiudb`fntf. T5< Qrgprjfcdcf cf tfrf` tfrf` td`dnmgs cfsjiudjs. Cf `gcg quf Q 4

vfrcdcfjrd . ⇕ T4, Q5 ⇕ T5 f d rfkjprgkd tdè³ff`` ³f vfrcdcfjrd.

;. [fld` ] 4 ]5 , Z4Z5 suekgnluntgs cg kgnluntg unjvfrsg V. [upgn [upgnmd md quf quf ] 4 ]5 0 V f Qrgvf vf quf quf ] 4 0 Z4 f ]5 0 Z5 . quf ] 4 Z4 f quf ] 5 S5 . Qrg Z4 Z5 0 , quf

∪

∆

⊅

∬

⊅

[gbu½ k˒ dg< 4 Kg`g pg r mjp³ `gst `gstrdcg rdcg pgr quf< quf< ] 4gtfsf 0 Z] 4.

dg edstd prgvdr quf ] 4 ⊋ Z4 quf pgr cupbd jnkbus˒dg tfrf`gs ⊅ Z4 fnt˒dg Qdrd `gstrdr quf ] 4 ⊋ Z 4 tg`f`gs tg`f`gs u u` ` fbf`fntg fbf`fntg y ∍ Z 4 . Kg`g pgr mjp³gtfsf mjp³gtfsf ] 4 ∬ ] 5 0 V

fnt˒dg dg y pfrtfnkf d ] 4 gu pfrt pfrtfnkf fnkf d ] 5.

5

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

∍

⊅

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

∍

∪ Z 5 0 ∆. Bgig Bgig y ∍

[f [f y ] 5 f ]5 Z 5 fnt˒ ddg g y Z 5 . G quf sfrj sfrjdd u` desurc desurcgg l³d quf Z 4 ]4 g quf prgvd quf ] 4 Z 4 . F pgrtdntg pgrtdntg quf ] 4 0 Z4 .

⊋

Dndbgid`fntf sf prgvd quf ]5 0 Z5 .

3. Kg`pdrf Kg`pdrf g fxfrk fxfrk³³Ĵkjg dntfrjgr dntfrjgr kg` g prj`fjrg prj`fjrg f` f` tfr`gs tfr`gs cf cf kbdrfzd kbdrfzd f sj`pbjkjcdcf sj`pbjkjcdcf cgs fnunkjdcgs. @gstrf quf qudbqufr u` cfbfs pgcf sfr rfsgbvjcg rfsgbvjcg pfbg gutrg. [gbu½ k˒ dg<

⊋

Qdrd prgvdr`gs quf ]4 0 Z4 , pgr fxf`pbg prfkjs³ prfkjs³dvd`gs dpfnds `gstrdr `gstrdr quf< ] 4 Z 4. Dssj` sf tg`dr`g tg`dr`gss u` fbf`f fbf`fntg ntg u cf cf   V, Q4 kg`g d prgprjfcdcf cf pfrtfnkfr d ] 4 f T4 kg`g d prgprjfcdcf cf pfrtfnkfr d Z 4 . Fnt˒dg pgcf`gs dfflr`dr quf Q 4 T 4 . L³d quf ] 4 Z 4 . Nfstf kdsg prgvdr d rfkjprgkd (T 4 Q 4), sfrjd g fqujvdbfnt fqujvdbfntff d prgvdr ]4 Z 4.

⇕ ⊋

⇕

⊅

F` gutrds pdbdvrds prgvdr d qufst˒dg dg ; j`pbjkd nd prgvd cd qufst˒dg 4 f vjkf-vfrsd.

⇕ T4 f Q5

6. Djncd ng tf`d cg prj`fjrg fxfrk fxfrk³³Ĵkjg, sfrjd v³dbjcg dbjcg suestjtujr ds j`pbjk j`pbjkd½ d½k˒ k˒gfs gfs Q 4

⇕ T5 nd mjp³ mj p³gtfsf pgr suds rfkjprgkds rfkj prgkds T 4 ⇕ Q4 f T5 ⇕ Q 57 [gbu½ k˒ dg<

⇕

⇕

Fssd suestjtuj½k˒ kddg ˒g n˒dg gerjid ger jid d j`pbj j `pbjkd½k˒ kd½k˒ dg dg Q4 T4 f Q5 T5. Edstd Edstd j` j`dijndr dijndr g fxf fxf`pbg `pbg f` quf V 0 N , Q4 f³ d prgprjfcdcf ‒n ³f pdr‐, Q5 sjinjfflkd ‒n ³f j`pdr‐, T4 quf cjzfr ‒n f `³ubtjpbg cf 3‐ f T 5 cjz ‒n ³f u` nu`frg prj`g `djgr cg cg quf 5‐.

8. Fskrfvd ds j`pbjkd½k˒ gfs gfs b³gij gijkds kds quf q uf kgrrfs kgrrfspgnc pgncff ` ad rfsgbu½ rfsgb u½k˒dg dg cd fqud½ fqu d½k˒dg dg

∜ x +5 05,vfld

qudjs s˒dg dg rfvfrs³ĴĴvfjs vfjs f fx fxpbjquf pbjquf g dpdrfkj`fntg dpdrfkj `fntg cf rd³Ĵzfs Ĵzfs fstrdnmds. fs trdnmds. Hd½ kd g `fs`g kg` d fqud½k˒ fqud½k˒ dg dg x + ; 0 x .

∜

[gbu½ k˒ dg< Hdzfnc Hdzfncgg y 0

∜ x tˈf` sf< sf< ⇕ y + 5 0 y 5 ∙

⇕ y5 ∕ y ∕ 5 0 1 5)(y +4)0 1 (4) ⇕ (y ∕ 5)(y ⇕ y 0 5, y 0 ∕4 (5) ∜   kg`g   y 0 x cf (4) f (5) tf kg`g tf`gs `gs 1, d fqud½k˒ddgg

u`d rdjz. ∜ x + ` 0 x tf` fxdtd`fntf u`d

[gbu½ k˒ dg< [fld   y 0 [fld

∜ x fnt˒dg< dg<

∜ x + ` 0 x pgcf sfr fskrjtd kg`g y + ` 0 y 5 ⇕ y5 ∕ y ∕ ` 0 1 Dpbjkdncg emdsodrd<

∜   ∕ (∕4) ´ (∕4)5 ∕ 3(4)(∕`) 4 ´ 4 + 3` y 0 0



5(4)

5

5

y ` 0 1 pgssujr³d cuds Kg`g pgr mjp³gtfsf ` > 1 fnt˒ fnt ˒dg dg d (4 + 3 `) > 1 f d fqud½k˒ddg g y rd³Ĵzfs, Ĵzfs, u`d pgsjtj pgsjtjvd vd f u`d nfidtjvd quf kmd`drf kmd`drf`gs `gs cf cf o 4 f o5 (kg` (kg` o 4 , o5 U ).

∜   [f [f y 0 ∕o5 fnt˒ dg dg x 0 ∕o5 f x 0 (o5 )5 sfncg dssj`< 3

∕

∕ ∕

∍

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

∜ x + ` 0 x ⇕ ∕o5 + ` 0 (o5)5 ⇕ ` 0 (o5)5 + o5 ⇕ ` > (o5)5 (4) ∜   dg< dg< `ds kg`g x 0 ∕o5 fnt˒ ` > (o5 )5

⇕ ` >x

Fssd j`pbjkd j`pbjkd½½k˒dg dg ng fntdntg ³f u` desurcg pgjs pgj s dndbjsdncg dndbj sdncg d fqud½ fqu d½k˒ddgg ( x + ` 0 x),f d kgncj½k˒ kd˒dgg cf quf ` > 1 fnt˒ fnt ˒dg dg cfvf`g cf vf`gss tfr x `. Bgig x n˒ddgg pgcf ssfr fr jiudb jiudb d o5 pgjs jssg rfsubtdrjd f` f` ` > x

∜   ∕

≦

Bgig d fqud½ fqu d½k˒ddgg s³g pgssuj u`d rdjz. F fbd ³f pgsjtjvd.

=. Kgnsjcfrf ds sfiujntfs (d (dpdrfntfs) pdrfntfs) fqujvdbˈ fnkjds fnkjds b³gijkds< gijkds< x 0 4

⇓ x5 ∕ 5x + 4 0 1

⇓ x5 ∕ 5 ¹ 4 +4 01 ⇓ x5 ∕ 4 0 1 ⇓ x 0 ´4 Kgnkbus˒ ddgg (7)< x 0 4

⇓ x 0 ´4. Gncf fst³d g frrg7

[gbu½ k˒ dg< G prgebf`d fst³d nd sfiuncd j`pbjkd j`pbjkd½½k˒dg. d g. Fn Fnqu qudn dntg tg   x5 5x + 4 0 1 x5 5 4 + 4 0 1 d 5 rfkjprgkd rfkjprg kd n˒ddgg ³f vf vfrcdcfjr rcdcfjrd, d, pgjs sf sf   x 0 4 fn fnt˒ t˒dg ( 4) 5 4 + 4 tdè³ tdè³ ff` ` ³f jiudb jiudb d zfrg. zfrg. Gu jrrfvfrs³ Ĵvfb. Ĵvfb. pdrd x 5 5(4) + 4 0 1 ³f jrrfvfrs³ sfld, d pdssd pdssdif` if` cf x 5 5x + 4 0 1 pdrd

∕

∕

∕ ∕ ∕ ¹

∕

⇕ ∕ ¹

:. Ds rd³Ĵzfs Ĵzfs cg pgbjnˈg`jg g`jg x ; 8x5 + 44 44x x 8 s˒ddgg 4, 5 f ;. [uestjtud [uestjtud,, nfssf pgbjnˈg`j pgbjnˈg`jg, g, g tf tfr`g r`g 44x 44x pgr 44 5 0 55, getfncg fnt˒dg dg   x; 8x5 + 48, quf djncd tf` 5 kg kg`g `g rdjz rdjz `ds n˒ dg sf dnubd dnubd pdrd   x 0 4 nf` pdrd nf` x 0 ;. Fnu Fnunkjf nkjf u` rfsubtdcg rfsubtdcg ifrdb ifrdb quf fxpbjquf fxpbjquf fstf hdtg f g rfbdkjgnf rfbdkjgnf kg` g fxfrk³ fxf rk³Ĵkjg Ĵkj g dntfr dntfrjgr jgr..

¹

∕

∕

∕

[gbu½ k˒ dg< rdjz jz ³f δ fnt˒ dg dg p(δ p( δ) 0 1. Sg`dnc Sg`dncgg digrd digrd Cdcg u` pgbjnˈg`jg p(x) 0 dx ; + ex5 + kx + c kuld rd u` sfiuncg pg pgbjnˈ bjnˈg`jg q(x) 0 kx pgcf-sf fskrfvfr fskrfvfr p(x) kg`g< p(x) 0 dx; + ex5 + q (x) + c

6

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

Ufpdrf Ufpd rf qu quff fst³d suestjtuj suest jtuj½½k˒ddgg ndcd ùcd ùcd f` p(x p(x). ). Cf `gcg `gcg qu f p p(( δ) kgntjnud kgntjnud sfncg sfncg rdjz cf p(x). p(x). Cfssf Cfssf `g `gcg cg suestjtuj suestjtujrr q( δ) pfbg tf tfr`g r`g ‒qx‐ ‒qx‐ f` p(x) sjinjfflkd sjinjfflkd dpfnds dpfnds quf fstd fstd`gs `gs suestjtujn suest jtujncg cg x pgr δ f` ‒qx‐. ‒qx‐. p( p(x) x) 0 dx ; + ex5 + kδ + c Dssj`   δ kgntjnud sfncg rdjz cf p(x) Dssj` p(x),, `ds ds suds cf`djs rd³Ĵzfs Ĵzfs pfrcf` g sfntjcg. Fssf hdtg tdè³ f` f` sf vfrjffl vfrjfflkd kd ng ng fxfrk³ fxfrkĴkjg Ĵ³kjg dntfrjgr qudncg suestjtu³ suestjtu³Ĵ`gs x pgr 4 nd fqud½ k˒ dg dg 5

x

∕ 5x + 4 0 1 41. [fld Q( Q(x) x) u`d kgncj½ kgncj½k˒ k˒ dg dg fnvgbvfncg d vdrj³dvfb d vfb x. (4) ‒Qdrd ttgcg gcg x, ³f sdtjshfjtd sdtjshfjtd d kgncj½kk˒ dg d˒g Q(x)‐ (5) ‒Fxjstf ‒Fxjstf dbiu dbiu` ` x quf sdtjsh sdtjshdz dz d kgncj½ kgncj½ k˒ dg dg Q(x).

d) [fncg D g kgnlu kgnluntg ntg cf gelft gelftgs gs x (cf u` kfrt kfrtgg kgnluntg kgnluntg unjvfrsg unjvfrsg V) quf sdtjsh sdtjshdzf dzf` `d kgncj½k˒ kd˒dgg Q(x), fskrfvd ds sfntfn sfntfn½½kds kds (4) (4) f (5) dkj`d, usdncg d bjniudif` cgs kgnluntgs. kgnluntgs. e) Tudjs ds nfid½k˒ gfs gfs cf (4) f (5)7 k) Qdrd kdcd sfntfn½kd dedjxg cjid sf fbd ³f vfrcdcfjrd gu hdbsd f hgr`f sud nfid½k˒ kddg< ˒g< 5

Fxjstf u` nu`f nu`frg rg rfdb rfdb   x tdb quf quf x 5 0 ∕4. ••    Qdrd tgcg nu`frg jntfjrg jntfjrg n , vd vdbf bf   n > n. • Qdrd tgcg nu`frg rfdb x , tf`-sf tf`-sf   x > 4 gu gu   x 5 2 4. ndturdb n tdb quf quf n > x. • Qdrd tgcg nu`frg rfdb x fxjstf u` nu`frg ndturdb • Fxjstf u` nu`f nu`frg rg ndtur ndturdb db   n tdb quf, pdrd tgcg tgcg nu`frg nu`frg rfdb x , tf tf` ` sf n > x. [gbu½ k˒ k˒dg cf D<

∍

Cd sfntfn½kd kd (4) kgnkbuj-sf kgnkbuj-sf dpfnds dpfnds quf< tgcg x V tdè³ ff` ` pfr pfrtfn tfnkf kf d D. D. N˒ ddgg sf pgcf cjzfr quf D 0 V pgrquf pgrq uf n˒dg sf sdef sf V f³ kgnstjtu³Ĵcg Ĵcg dpfnds cf gelftgs x. Cf (5) sf kgnkbu kgnkbuj-quf j-quf D 0

 ∆

Dssj` ds sfntfn½ sfntfn½ kkds ds (4) f (5) fskrjtds nd hhgr`d gr`d cf kgnluntg kgnluntg sfrjd` rfspfktjvd`fntf< (4)D 0

{x|x ∍ V} (5) (5) D  0∆

[gbu½ k˒ k˒dg cf E< Dn nfid fid½½k˒ k˒dg dg cf cf (4) ³ff 4 D nfid nf id½ ½k˒ k˒ dg dg cd dfflr` dfflr`d½ d½k˒ d dg g sfr³ sfrd< d r . D nfid½ nf id½k˒ dg d g cd cd dffl dfflr`d r`d½ k½˒ dg sfr³ d r. D nfid nf id½ ½k˒ k˒ d dg g cd dffl dfflr` r`d½ d½k˒dg dg sfr³ sf r³d d 1 x∕6 5x + 8 > 1 x 6

∕

Kuld cfsjiudbcdcf gkgrrf pdrd x > 6 f x 2 ;. L³d d sfiuncd jnfqud½k˒dg dg hdrf`gs dssj`< x x

∕4 ∕ 6 2 4

;5

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

∕ ¹ xx ∕∕ 64 2 4 ¹ (∕4)

( 4)

4 x 4

∕ x > ∕4 ∕6

x

+

x

6

> 1

∕ ∕ x∕6 x∕6 4 ∕ x + (x ∕ 6) > 1 x∕6 4

∕ x + x ∕ 6 > 1 x∕6 ∕3 > 1 ∕6

x

Kuld Kul d sgb sgbu½ u½k˒ddgg gkgrrf sg`fntf pdrd x 2 6 (edstd (edstd gbmdr prg cfng` cfng`jndcgr). jndcgr). Dssj` hdzfncg d jntfrkfss˒ dg dg fntrf ds sgbu½ sgbu ½k˒gfs gfs fnkgntrd`gs kg`g sgbu½kk˒ dg d˒g d kgncj k gncj½½k˒dg dg cf quf x 2 ;. c) Nfssf kdsg prgkfcf`gs cd sfiujnt sfiujntff hgr`d<

|x ∕ 5| + |x + 3| 0 x∕(∕x5∕+5)|x++|x3|+03|=0 =



Cf kdcd kdc d fqud½ fqud ½k˒dg dg dkj`d djncd tf`-sf< x

∕ 5 + |x + 3 | 0



x x

∕ 5 + x +3 05 x +5 0= ∕ 5 ∕ (x + 3 ) 0 ∕8 0 =

f tdè³ f`< f`<

5

∕ x + |x + 3 | 0



5 5

∕ x + x + 3 0 8 0 = ∕ x ∕ x ∕ 3 0 ∕5x ∕ 5 0 =

Cgs cgjs ³ubtj`gs sjstf sjstf`ds `ds pfrkfef`gs quf ds ³unjkds ³unjkd s sgbu½kk˒gfs ˒gfs pgss³Ĵvfjs Ĵvfjs vˈff` ` cf 5x + 5 0 = x 0 ; f c cff 5x 5 0 = x 0 6.

∕ ∕

⇕

∕

Cf hdtg tfstdncg fstfs vdbgrfs tf`gs<

|(∕6) ∕ 5| + |(∕6 ) + 3| 0 | ∕ 9| + | ∕ 4| 0 = |(;) ∕ 5| + |(;)+3 | 0 |4| + |9| 0 = ;;

⇕

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

Dssj` d sgbu½k˒ sgbu½k˒ ddgg pdrd d fqud½k˒ fqud½k˒dg dg sfrjd 1.





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Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

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85

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Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

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D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

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D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

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98

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Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

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Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

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Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

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;

[gbu½ k˒ dg< G `³ ftgcg ftgcg cf Nfwtgn ³f u` `³ fftgcg tgcg nu`³ frjkg frjkg pdrd cftfr`jndr ds rd³Ĵzfs Ĵzfs rfdjs rfdjs cf u` pgbjnˈ g`jg. Nfstf kdsg cg pgbjnˈg`jg p (x) 0 x; 5 (vfld K³dbkubg kg` k g` ifg`ftrjd dndb³Ĵtjkd cg Bgujs Bftmgbc, vgbu`f 4, p³dijnd 84). Kg`f½kdncg kdncg d pdrtjr cf x 1 0 4 getf getf`gs< `gs<

∕

x4 0 4.;;; x5 0 4.58;: x; 0 4.56::: Kg`g (4. (4 .56:::); 2 5 2 (4 (4..58); d dprgxj`d½kk˒dg ˒dg pdrd

∜ 5 kg`g kg `g pfcj pfcjcg cg ³f cf 4.56::. ;

[f dbiu`d dbiu`d pdssdif` pdssdif` fflkgu gesku rd gu sfr sf dbiu` dbiu hgj hgj kgrrf½ kg`ftjcg kg`f pgr hdvgr fskrfvd fskrfvd pdrd njeebfcjfigAi`djb.kg` pdrdgeskurd quf pgssd hfjtg`d frrg cfvjcd kgrr f½tjcg k˒dg. dg.pgr Qdrd fnkgntrdr fnkg ntrdr fssf f gutrgs fx fxfrk frk³³Ĵkjgs rfsgbvjcgs rf sgbvjcgs c cff `dtf`³ dtjkd dtjkd dkfssf< www.nuèfr.=:1`.kg`

=8

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

³ JKD ³ D @DSF @DSF@ @DS DSJK D CG FN[J FN[JNG NG @ FCJG D `dtf`³dtjkd cg F nsjng `³fcjg fcjg (vgbu`f 4) Fbgn Bdifs Bj Bj`d `d Qdubg Kfzdr Qjntg Kdrvdbmg. Fcudrcg Ydinfr. Duiustg K³ffsdr sdr @gridcg. Ufsgbvjcg Ufsgb vjcg pgr< Cjfig Gbjvfj Gbjvfjrd rd

=

Hun Hun½kgfs gfs Fxpgnfn ˒ Fxpgnfnkjd kjdss f Bgidr Bgidr³ ³Ĵt`jkd Ĵt`jkdss

4. Kg` u` b³dpjs b³dpjs kuld pgntd tf` 1,15 `` cf fspfssurd, fspfssurd, cfsfld-sf trd½ kdr g ir³dfflkg cd hun½k˒ dg dg h   (x) 0 5x . Dtf³ quf cjstˈ ddnkjd nkjd ³d fsqufrcd cg fjxg vfrtjkdb pgcf-sf jr sf` quf g ir³ dfflkg dtjnld g fjxg mgrjzgntdb7 [gbu½ k˒ dg< Kmd`dncg cf cf r g rdjg cd pgntd cg b³dpjs, b³dpjs, fnt˒ ddgg d bj bjnmd nmd quf fseg½ f seg½kd kd g ir ir³³dfflkg tgkdr³d g fjxg G] ng pgntg ( x, 5x )kg`5 x 2 r . Ufsgbvfncg d jnfqud½kk˒ dg d˒g hgr`dcd getf`gs d sgbu½k˒dg. dg. 5x 2 r kg` kg` ( r > 1) x

bgi(5 (5 ) 2 bgi bgi (r ) ⇕ bgi bgi(5) (5)   2 bgi bgi (r ) ⇕ x ¹ bgi bgi bgi((r) ⇕ x 2 bgi bgi(5) (5) kg`g   bgi (5) kg`g

bgi( bgi (r ) ≍ 1.;14 fn fnt˒ t˒dg dg   x 2 . 1.;14

Dssj`, g ir³dfflkg tgkdr³d g fjxg mgrjzgntdb ng pgntg gncf d deskjsd ³f j`fcjdtd`fntf `fngs quf bgi bgi((r ) . 1.;14

5. Cfˈ fxf`pbg cf u`d hun½kk˒dg ˒dg krfskfntf krfskfntf h   < U   U + tdb quf quf,, pdr pdrdd tgcg tgcg x U , d ssfq fquˈ uˈ fnkjd fnkjd h   (x + 4), 4), h   (x + 5),...,h 5),...,h   (x + n),... f³ u`d prgirfss˒ dg dg ifg`³ ftrjkd ftrjkd `ds `ds h   n˒ ddgg ³f c cgg ttjp jpgg h   (x) 0 e dx .

↕

[gbu½ k˒ dg< Sg`dncg h   (x) 0 xe xe (hun½kk˒dg ˒dg bjnfdr) bjnfdr),, fnt˒ ddgg

=9

∍

¹

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

 

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED









h   (1 +4), +4), h   (1 + 5) 5),,

+4), h   (4 + 5) 5),, h   (4 +4),



  ¹¹¹

¹ ¹ ¹ , h   (1 + n), ¹ ¹ ¹ 0 (e,e, ¹ ¹ ¹ 

¹ ¹ ¹ , h   (4 + n), ¹ ¹ ¹ 0 (e,e, .. .







h   (n + 4) 4),, h   (n + 5) 5),,



¹ ¹ ¹ , h   (n + n), ¹ ¹ ¹ 0 ( e,e, ¹ ¹ ¹



.. . Tuf s˒dg prgir p rgirfss˒ fss˒gfs gfs ifg`³fftrjkds trjkds kgnstdn kgnstdntfs tfs (Q.I. cf rdz˒ dg ji jiudb udb d 4) 4).. Kg`g Kg`g   h   (x) n˒dg dg ³f cg tjpg tjpg e dx fnt˒ddgg ³f u`d rfspgs rfspgstd td dkfjt³ dk fjt³ dvfb dvfb dg prgebf`d prgebf`d..

¹

;. Cd Cdcg cgss d > 1 f e > 1, dègs dègs cjhfr cjhfrfntf fntfss cf 4, qudb d prgprjfcd prgprjfcdcf cf cd hun½ hun½ k˒ ddgg fxpg fxpgnfnkj nfnkjdb db quf dssfiurd d fxjstˈfnkjd fnkjd cf m 0 00 0 1 tdb tdb quf quf e x 0 dx/m pdrd tgcg x U 7 @gstrf kg`g getfr g ir³dfflkg x cf cf y 0 e x d pdrtjr pdrtj r cg ir³dfflkg cf y 0 dx . Vsf ssud ud kg kgnkbus˒ nkbus˒dg dg pdrd trd½kdr g ir³dfflkg cf y 0 4/ 3 x d pdrtjr pdrtj r cg ir³dfflkg cf y 0 5 .



∍

 ∜    ;

[gbu½ k k˒ ˒ d dg g cd prj`fjrd pdrtf< U

+

↕ U

x

D prgprjfcdcf f`ntg, qufst˒dg qufst˒ dg cjz qu hun½ k˒ dg dgtdb fxpgnfnkjdb fxpgnfnkjdb cffflnjcd jcd. pgr pCgrdĴ³ h e x(x0 )0 e , ³f sgerfl sgerflftjvd ftjvd.. Qgrtdntg, Qgrtd cdcg cdcg d >quf 1,f d mhun½k˒ b quf quf em 0 dh , 1 qudjsqufr, `gstrf quf fxjstf d > 1 tdb tdb quf d x 0 y1 .



1

[gbu½ k˒ dg< 4 x Sg`dncg d 0 y1 1 fnt˒ ddg g dx 0 1

x1

4 x y 1

 

0 y1 , kg`g rfqufrjcg. rfqufrjcg.

1

dg-nubgs f cf `fs`g sjndb, prgvf quf fxjstf` d > 1 f e tdjs quf 8. Cdc Cdcgs gs x 1 0 x 4 f y1, y4 n˒dg-nubgs e dx 0 y1 f e dx 0 y 4.

¹

1

 ¹

4

[gbu½ k˒ dg< Edstd tg`dr tg`dr d 0

 y1 y4

4 x1 ∕x4

f e 0

y1 . dx 1

9. D iirdnc rdncfzd fzd y sf fxprj`f kg`g y 0 e dt f` hun½k˒dg dg cg tf`pg t. [f [fld ld` ` c g dkr³ ffskj`g skj`g quf

¹

sf cfvf cdr d t pdrd quf y cgerf f ` (`fjd-vjcd t/c cf y ) g dkr cf dkr³ fskj`g f³skj`g cf t nfkfss³drjg drjg pdrd quf y sf rfcuzd ³d `ftdcf. @gstrf quf ` 0 c f y 0 e 5 , bg bgig ig   c 0 bgid 5 0 4 /bgi5 d.

∕

¹

:1

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

[gbu½ k˒ dg< Xd`gs kg`f½ kdr prgvdncg quf sf sf y (t + c) 0 5w f y (t) 0 w , kg kg` ` y (t)edt , fnt˒dg dg c 0 Qrgvd. y(t + c) 0 5w edt+c 0 5w

⇕ edt+c 0 ∜ 3 w ⇕ Kg`g   ed t 0 w fnt˒ Kg`g dg< dg<

∜ 3 w ⇕ dc 0 ∜ 3

wdc 0

4 ⇕ c 0 bgi bgi((d) 0 bgi(3) bgi(3) 5 bgi(3) (3) ⇕ c 0 5 bgi ¹ bgi( bgi(d) K.T. C. Qrgvdcd d dfflr`d½kk˒ dg d˒g pdrtjrf`gs digrd pdrd d rfsgbu½k˒dg dg cg prgebf`d prgpgstg. [fld   y (t) 0 w kgnsjcfrdncg g fnunkjdcg tf`gs< [fld y(t + c) 0 edt+c 0 5w (4) 4 y (t + `) 0 edt+` 0 edt+` 0 w (5) 5 Kg`pdrdncg Kg`pd rdncg (4) (4) kg` (5)

¹

y(t + c) 0 3 y(t + `)

⇕ edt+c 0 3 ¹ edt+` ⇕ edtdc 0 3 ¹ edt d` c

⇕d

`

0 3d c

⇕ dd` 0 3 :4

bgi(3) bgi(3) . 5 bgi( bgi(d)

¹

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

⇕ dc

`

∕

03

Dpbjkdncg bgidrjt`g bgi dc∕` 0 bgi(3) bgi (3)

 

bgi(d) 0 bgi(3) bgi (3) ⇕ (c ∕ `)bgi( bgi bgi(3) (3) ⇕ c ∕ ` 0 bgi( bgi(d)

V`d vfz quf prgvd`gs quf c 0

bgi(3) bgi(3) fnt˒ ddgg 5 bgi( bgi(d)

¹

c

bgi(3) bgi(3) ∕ ` 0 bgi bgi((d)

bgi bgi(3) (3) ⇕ ` 0 c ∕ bgi( bgi(d) bgi(3) bgi(3) bgi (3) ⇕ ` 0 5 bgi(3) ∕ bgi( ¹ bgi( bgi (d) bgi (d) bgi (3) 0 ∕c ⇕ ` 0 ∕ 5 bgi(3) bgi( bgi (d) ¹ ⇕ ` 0 ∕c Kg` jssg fflkd prgvdcg quf ` 0 ∕c, f ùcdncg ùcdncg d edsf edsf cf c cf 41 pdrd 5 kgnkbu³Ĵ-sf Ĵ-sf tdè³ f` f`

quf<

c 0

bgi(3) bgi(3) 4 0 5 bgi( bgi (d) bgi5 (d)

¹

K.T. C.

=. Ges Gesfrv frvd½ d½k˒ kg˒gfs fs hfjtds curdnt curdntff bgnig tf`pg `gstrd` quf, dp³gs pfr³Ĵgcg Ĵgcg cf `fs`d curd½k˒ kdg, dg ˒ ,d pgp pgpub ubd½ d½k˒ k˒ddgg cd ttfrrd frrd fflkd ùbtjpbjkdcd ùbtjpbjkdcd pfbg `fs`g `fs`g hdtgr. [defncg quf fssd pgpubd½ k˒ dg dg frd cf 5,8= ejbm˒ gfs gfs f` ;,9= ej ejbm˒ bm˒D gfs f` 4:95 4:95, pfcf-sf< sf< (d) G tf`pg tf`p nfkfss³drjg ³drjg qufddpgpubd pg pubd½ dg ddg cd tfrrd cge4:68 cgerf rf cffvdbgr? (e) pgpubd½k˒ dg d,gpfcffstj`dcd fstj`dcd pdrd pdrd g dngg5145? 5nfkfss 145? (k) F` pdrd quf dng ppgpubd½ gpubd½ ½ k˒ k˒ dgg cd tfrrd frd cf 4 ejbm˒dg.

:5

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

[gbu½ k˒ k˒dg cf d< Dp³ gs gs u` tf`pg ‒t‐ d pgpubd½ pgpubd½k˒ k˒ dg dg ³f u`d u` d fx fxprf prfss˒ ss˒ dg dg cg tjpg y (t) 0 e fdt gncf ‒e‐ ³f d pgpub pg pubd½ d½ k˒ kdg d˒g jnjkjdb (5.8= `jbm˒gfs). [fncg dssj`

¹

y(t) 0 ;.9= d(4:95∕4:68)

0 ;.9= ⇕ 5.8= ¹ f ⇕ d ≍ 1.1546

⇕ 5.8= ¹ fdt 0 5 .8= ¹ f1.1546t Tudncg d pgpubd pgpubd½½k˒dg dg cd tfrrd cgerdr tfrf`gs< y(t) 0 5e

⇕ eeff1.1546t 0 5 e ⇕ f1.1546t 0 5 ⇕ t ≍ 5; 5;,, 53 [gbu½ k˒ kd ˒ dg< g< 5;,53 dngs.

[gbu½ k˒ k˒dg cf e< F` 5145 tfrf`gs tfrf`gs t 0 68 ((5145 5145

4:688 0 45) 45),, dssj` dssj` ∕ 4:6

5.8= 8=ff1.1546¹45 0 = .:ej

[gbu½ k˒ kd ˒ dg< g< D pgpub p gpubd½ d½k˒dg dg cd tfrrd ssfr³ fr³d cf =.: ejbm˒gfs.

[gbu½ k˒ k˒dg cf k< 5.8= 8=ff1.1546t 0 4

⇕ 1.1546 1546tt ¹ bn bn((f) 0 bn bn(4 (4//5.8=) t

≍ ∕36 36..=9

∕

Hjndb`fntf Hjndb` fntf,, hdzfncg hdzfncg 4:68 4:68 + ( 36 36..=9) kmfid`gs d sgbu½ sgbu½k˒ k˒ dg. dg.

∕

4:68 4:68 + ( 36. 36.=9) =9) 0 4:68 4:68 0 4:41 4:41..4;

∕ 36 36..=9

:;

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

Dssj`. cfskgerf-sf qu quff g gkgrrjc gkgrrjcgg hgj ng dng cf 4:41.

:. Cˈf u` driu`fntg jncfpfncfn jncfpfncfntf tf cf gesfrvd½ k˒ gfs gfs pdrd lustjfflkdr lustjfflkdr quf, f` kgncj½k˒ kgncj½k˒ gfs gfs ngr`djs, d pgpub p gpubd½ d½k˒dg dg cd tfrrd dp³gs g cfkursg cf kursg cf pfr³Ĵgcgs Ĵgcgs jiudjs fflkd ùbtjpbjkdcd ùbtjpb jkdcd pfbd `fs`d kgnstdntf. [gbu½ k˒ dg< 777

41. Ufsgbvd gs fxfrk³Ĵkjgs Ĵkjgs cg bjvrg ‒Bgidrjt`gs‐, ‒Bgidrjt`gs‐, fspfkjdb`fntf gs cg ³u ubtj`g btj`g kdp³ Ĵtubg. Ĵtubg. [gbu½ k˒ dg< G prghfssgr prghfss gr quf kgerdr cg dbung fssd qufst˒ dg ³f t˒ dg sf` ng½k˒dg dg quf `frfkf u`d surrd!

[f dbiu`d dbiu`d pdssdif` pdssdif` fflkgu gesku rd gu sfr sf dbiu` dbiu hgj hgj kgrrf½ kg`ftjcg kg`f pgr hdvgr fskrfvd fskrfvd pdrd njeebfcjfigAi`djb.kg` pdrdgeskurd quf pgssd hfjtg`d frrg cfvjcd kgrr f½tjcg k˒dg. dg.pgr Qdrd fnkgntrdr fnkg ntrdr fssf f gutrgs fx fxfrk frk³³Ĵkjgs rfsgbvjcgs rf sgbvjcgs c cff `dtf`³ dtjkd dtjkd dkfssf< www.nuèfr.=:1`.kg`

:3

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

³ JKD ³ D @DSF @DSF@ @DS DSJK D CG FN[J FN[JNG NG @ FCJG D `dtf`³dtjkd cg F nsjng `³fcjg fcjg (vgbu`f 4) Fbgn Bdifs Bj Bj`d `d Qdubg Kfzdr Qjntg Kdrvdbmg. Fcudrcg Ydinfr. Duiustg K³ffsdr sdr @gridcg. Ufsgbvjcg Ufsgb vjcg pgr< Cjfig Gbjvfj Gbjvfjrd rd

:

Hun Hun½kgfs gfs Srjign ˒ rjigng`³ g`³ ftrjkds ftrjkds

4. Cftfr`jnf gs vdbgrfs `³dxj`g f `³Ĵnj`g cd hun½kk˒ ddg ˒g h   < ;/(5 + sfn sfn((x)).

↕ U      cffflnjcd pgr h   (x)

U

0

[gbu½ k˒ dg< Kg`g 4 sfn sfn((x) 4 fnt˒dg h(x) ³f `³ dxj`d dxj`d qudncg sfn sfn((x) 0 4 (gu (gu sfl sfld, d, qud qudncg ncg g cfng`jndcgrr cf h(x) ³f `³Ĵnj`g) cfng`jndcg Ĵnj`g) f `³Ĵnj`d Ĵnj`d qudncg sfn sfn((x) 0 4 (qudn (qudncg cg g cfng`jndcgr cfng`jndcgr cf h(x) ³f `³ dxj`g). dxj`g).

∕ ≣

X`dx 0

≣

∕

; 0; 5 + ( 4)

∕

X`jn 0 ; 0 4 5 +(4)

5. Gesfr Gesfrvdncg vdncg d ffli ffliurd urd d sfiujr, sfiujr, gncf DE 0 x, `gst `gstrf rf quf quf t 0 sfn sfn((x)/kgs /kgs((x). t 4 E x

[gbu½ k˒ dg< Srdkf u`d rftd kg`g d rftd pgntjbmdcd nd ffliurd d sfiujr. :6

G D

D @dtf`³dtjkd cg Fnsjng @³ffcjg cjg

Cjfig Gbjvfjrd - Xjt³ grjd cd Kgnqujstd Kgnquj std / ED

t E x G

K

D

∲ ∈ SGD ddssj`< ssj`<

Gesfrvf quf ∈ GEK

KE GK   0 DS   GD

sfn((x) kgs( kgs(x) sfn sfn((x) ⇕ sfn 0 ⇕ t 0 t t kgs kgs((x)

¹

;. [f sfn sfn((x) + kgs( kgs(x) 0 4 .5, qudb ³f g vdbgr cg prgcutg sfn sfn((x) kgs kgs((x)7 [gbu½ k˒ dg< sf sfn( n(x) x) + kgs( kgs(x) x) 0 4. 4.55

⇕ (sfn(x)+kgs(x))5 0 4.4.33 33 4+5sfn(x) n(x)kgs(x kgs(x)) 0 4.33 ⇕ 4+5sf 5sfn(x)kgs(x)0 gs(x)0 1.33 ⇕ 5sfn(x)k 1.33 sfn(x)kgs(x) (x) 0 ⇕ sfn(x)kgs 5 ⇕ sfn sfn(x)k (x)kgs(x)0 gs(x)0 1.55 3) Cffflnj`gs Cffflnj`gs dquj ds hun½k˒ hun½k˒ ggfs< fs< sfkdntf

A Matemática Do Ensino Médio Vol 1

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