application of derivatives theory_e

7/21/2019 Application of Derivatives Theory_e

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DFXNW

"dfèsnjudfrpnysegs.e`" 6

FRRIEGFXEL@ LC HA_ESFXESAW

_fta lc gnf`ma, Xf`ma`t f`h @lrdfi, Arrlr f`h Fpprlxedftel`, _liia―s Xnalrad, IDSX, Dl`ltlègety,

Dfxedf Deèdf.

Harevfteva fs rfta lc gnf`ma

E` vfrelus ceaihs lc fppieah dftnadftegs là nfs tna quast tl j`lw tna rfta ft wnegn là vfrefoia esgnf`me`m, wetn raspagt tl ltnar. Xna rfta lc gnf`ma `fturfiiy racars tl teda. Out wa gf` nfva rfta lc

gnf`ma wetn raspagt tl ltnar vfrefoias fisl.

F` agl`ldest dfy wf`t tl stuhy nlw tna e`vas tda`t gnf`mas wetn raspagt tl vfreftel`s e` e`taras t

rftas.

F pnysegef` dfy wf`t tl j`lw, nlw sdfii gnf`mas e` hlsfma gf` fccagt tna olhy's raspl`sa tl f hrum.

F pnyse gest dfy wf`t tl j`lw tna rfta lc gnfr ma lc hestf`ga wetn raspagt tl teda.

Fii quastel`s lc tna folv a typa g f` oa e `tarpratah f`h raprasa`tah use`m harevftevas .

Hac e` et el` ; Xna fvarfma rfta lc gnf`ma lc f cu`gtel` c(x) wetn raspagt tl x lvar f` e`tarvfi Zf, f + nT es

haceàh fsn

)f(c )nf(c .

Hac e` et el` ; Xna e`stf`tfàlus rfta lc gnf`ma lc c(x) wetn raspagt tl x es haceàh fs c (x) 42n

ied n

)f(c )nf(c ,

prlvehah tna iedet axests.

@lta ; Xl usa tna wlrh 'e`stf tfàlus', x d fy `lt oa raprasa`te`m teda. ]a usufiiy usa tna wlrh 'rfta lc

gnf`ma' tl daf` 'e`stf`tfàlus rfta lc gnf`ma'.

Axfdpia # 6 Nlw cfst tna fraf lc f gergia e`grafsas wna` ets rfheus es 1gd:

(e) wetn raspagt tl rfheus (ee) wetn raspagt tl hefdatar

Wliutel` ; (e) F 4 r ? ,hr

hF4 ?r

1r hr

hF

4 62 gd ?/gd.

(ee) F 48

H? ,

hH

hF4

?

H

62HhH

hF

4

?

. 62 4 1 gd?/gd.

Axfdpia # ? Ec fraf lc gergia e grafsas ft f rfta lc ?gd?/sag, tna` ce`h tna rfta ft wnegn fraf lc tna e`sgreoah

squfra e`grafsas.

Wliutel` ; Fraf lc gergia, F6 4 r ?. Fraf lc squfra, F

? 4 ?r ? (saa cemura)



DFXNW

"dfèsnjudfrpnysegs.e`" ?

ht

hF64 ?r

ht

hr ,

ht

hF ?4 8r .

ht

hr

? 4 ?r .ht

hr r

ht

hr 4

6

ht

hF ?4 8 .

6

48

gd?/sag

Fraf lc squfra e`grafsas ft tna rfta8

gd ?/sag.

Axfdpia # 7 Xna vliuda lc f guoa es e`grafse`m ft f rfta lc = gd7/sag. Nlw cfst es tna surcfga fraf e`grafse m

wna` tna ia`mtn lc f` ahma es 8 gd<

Wliutel`. Iat ft slda teda t, tna ia`mtn lc ahma es x gd.

v 4 x7 ht

hv4 7x?

ht

hx(out

ht

hv4 =)

ht

hx

4 ?x7

=

gd/sag.

@lw W 4 3x?

ht

hW4 6?x

ht

hx

ht

hW4 6?x. ?x7

=4

x

?>

wna` x 4 8 gd,ht

hW4 = gd?/sag.

Axfdpia # 8 Wf`h es plure`m crld pepa ft tna rfta lc 6? gd7/s. Xna cfiie`m sf`h clrds f glà l` tna mrlu`h

e` sugn f wfy tnft tna naemnt lc tna glà es fiwfys là - sextn lc rfheus lc ofsa. Nlw cfst es tna

naemnt lc tna sf`h glà e`grafse`m wna` naemnt es 8 gd<

Wliutel`. S 47

6r ?n

out n 43

r

S 47

6 (3n)? n

S 4 6? n7

ht

hS4 73 n?.

ht

hn

wna`,ht

hS4 6? gd7/s f`h n 4 8 gd

ht

hn4 ?)8.(73

6?

4

8>

6gd/sag.

Waic prfgtega prloiads ;

(6) _fheus lc f gergia es e grafse`m ft rfta lc 7 gd/sag. Ce`h tna rfta ft wnegn tna fraf lc gergia es

e`grafse`m ft tna e`stf`t wna` rfheus es 62 gd.

(?) F ifhhar lc ia`mtn 1 d es iafè`m fmfe`st f wfii. Xna olttld lc ifhhar es oae`m puiiah fil`m tna



DFXNW


mrlu`h fwfy crld wfii ft rfta lc ?gd/sag. Nlw cfst es tna tlp pfrt lc ifhhar siehe`m l` tna wfii

wna` cllt lc ifhhar es 8 d fwfy crld wfii.

(7) ]ftar es hreppe`m lut lc f glègfi cu`ài lc sade-vartegfi f`mia 81² ft rfta lc ?gd7/s. Ce`h tna

rfta ft wnegn sif`t naemnt lc wftar es hagrafse`m wna` tna naemnt lc wftar es ? gd.

(8) F nlt fer ofiill` rese`m strfemnt up crld f iavai ceaih es trfgjah oy f rf`ma ce`har 122 ct crld tna

iect-lcc ple`t. Ft tna dlda`t tna rf`ma ce`har's aiavftel` f`mia es /8, tna f`mia es e`grafse`m ft

tna rfta lc 2.68 rfh/de . Nlw cfst es tna ofiill` rese`m ft tnft dlda`t.

F`swars ; (6) 32 g d?/sag (?)7

>gd /sag (7)

?

6gd/sag. (8) 682 ct/de`.

Xf`ma`t f`h @lrdf i

Iat y 4 c(x) oa cu`gtel` wetn mrfpn fs snlw` e` cemura. Gl`sehar sagf`t

RV. Ec V ta`hs tl R fil`m tna gurva pfsse`m tnrlumn tna ple`ts

V6, V

?, .....

E.a. V R, sagf`t RV weii oaglda tf`ma`t ft R. F ieà tnrlumn R

parpa`heguifr tl tf`ma`t es gfiiah `lrdfi ft R.

Maldatregfi Dafè`m lc

y

Fs V R, n 2 f`h silpa lc gnlrh RV ta`hs tl silpa lc tf`ma`t ft R (saa cemura).

Wilpa lc gnlrh RV 4n

)x(c )nx(c

RVied silpa lc gnlrh RV 4

2nied n

)x(c )nx(c

silpa lc tf`ma`t ft R 4 c (x) 4hx

hy

Aquftel` lc tf`ma`t f`h `lrdfi

)y,x( 66hx

hy

4 c (x

6) ha`ltas tna silpa lc tf`ma`t ft ple`t (x

6, y

6) l` tna gurva y 4 c(x). Na`ga tna aquftel`

lc tf`ma`t ft (x6, y

6) es meva` oy

(y ‗ y6) 4 c (x

6) (x ‗ x

6) : w na`, c (x

6) es rafi.

Fisl, se`g a `lrd fi es f ieà parpa`heguifr tl tf`ma`t ft (x6

, y6) sl ets aquftel` es meva` oy

(y ‗ y6) 4 ‗

)x(c

6

6 (x ‗ x

6), wna` c (x

6) es `l`zarl rafi.

Ec c (x6) 4 2, tna` tf`ma`t es tna ieà y 4 y

6 f`h `lrdfi es tna ieà x 4 x

6.

Ec 2n

ied n

)x(c )nx(c 66 4 lr ‗ , tna` x 4 x

6 es tf`ma`t (SA_XEGFI XF@MA@X) f`h y 4 y

6 es `lrdfi.

Axfdpia # 1 Ce`h aquftel` lc tf`ma`t tl y 4 ax ft x 4 2. Na`ga hrfw mrfpn

Wliutel` Ft x 4 2 y 4 a2 4 6



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Na`ga ple`t lc tf`ma`t es (2, 6)

hx

hy4 ax

2xhx

hy

4 6

Na`ga aquftel` lc tf`ma`t es

6 (x ‗ 2) 4 (y ‗ 6)

y 4 x + 6

Axfdpia # 3 Ce`h tna aquftel` lc fii strfemnt ieàs wnegn fra tf`ma`t tl gurva y 46x

6

f`h wnegn fra

pfrfiiai tl tna ieà x + y 4 2.

Wliutel` ; Wupplsa tna tf`ma`t es ft (x6, y

6) f`h et nfs silpa ‗ 6.

)y,x( 66

hx

hy4 ‗ 6.

‗ ?6 )6x(

6

4 ‗ 6.

x6

4 2 lr ?

y6

4 ‗ 6 lr 6

Na`ga tf`ma`t ft (2, ‗ 6) f`h (?, 6) fra tna raquerah ieàs (saa cemura) wetn aquftel`s

‗ 6(x ‗ 2) 4 (y + 6) f`h ‗ 6 (x ‗ ?) 4 (y ‗ 6)

x + y + 6 4 2 f`h y + x 4 7

Axfdpia # = Ce`h aquftel` lc `lrdfi tl tna gurva y 4 |x? ‗ | x | | ft x 4 ‗ ?.

Wliutel` ; E` tna àemnolrnllh lc x 4 ‗ ?, y 4 x? + x.

Na`ga tna ple`t lc gl`tfgt es (‗ ?, ?)

hxhy 4 ?x + 6

?xhxhy

4 ‗ 7.

Wl tna silpa lc `lrdfi ft (‗ ?, ?) es7

6.

Na`ga aquftel` lc `lrdfi es

7

6(x + ?) 4 y ‗ ?.

7y 4 x + >.

Axfdpia # > Rrlva tnft sud lc e targapts lc tna tf`ma`t ft f`y ple`t tl tna gurva x + y 4 f l` tnagllrhe`fta fxes es gl`stf`t.

Wliutel` ; Iat R(x6, y

6) oa f vfrefoia ple`t l` tna gurva x + y 4 f , fs snlw` e` cemura.

aquftel` lc tf`ma`t ft ple`t R es ‗6

6

x

y(x ‗ x

6) 4 (y ‗ y

6)

‗6x

x+ 6x 4

6y

y ‗ 6y



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6x

x+

6y

y4 6x + 6y

6x

x+

6y

y4 f ( fyx 66 )

Na`ga ple`t F es 2,fx6 f`h gllrhe`ftas lc ple`t O es )fy,2( 6 . Wud lc e`targaptss

4 f ( 6x + 6y ) 4 f . f 4 f (wnegn es gl`stf`t)

Axfdpia # 5 Ce`h tna aquftel` lc fii plsseoia `lrdfi/s tl tna pfrfolif x? 4 8y hrfw` crld ple`t (6, ?).

Wliutel` ; Iat ple`t V oa

8

n,n

?

l` pfrfolif x? 4 8y fs snlw` e` cemura

@lw, dRV

4 silpa lc `lrdfi ft V.

Wilpa lc `lrdfi 4 ‗nx

hy

hx

4 ‗n

?

6n

?8

n?

4 ‗

n

?

8

n7

‗ ?n 4 ‗ ?n + ?

n7 4 > n 4 ?

Na`ga gllrhe`ftas lc ple`t V es (?, 6) f`h sl aquftel` lc raquerah `lrdfi oagldas x + y 4 7.

@lta ; Xna aquftel` mevas lìy là rafi vfiua lc n, na`ga tnara es lìy là ple`t lc gl`tfgt edpiye`m tnft lìy

là rafi `lrdfi es plsseoia crld ple`t (6, ?).


(1) Ce`h tna silpa lc tna `lrdfi tl tna gurva x 4 6 ‗ f se` , y 4 o gls? ft 4?

.

(3) Ce`h tna aquftel` lc tna tf`ma`t f`h `lrdfi tl tna meva` gurvas ft tna meva` ple`ts.

(e) y 4 x8 ‗ 3x7 + 67x? ‗ 62x + 1 ft (6, 7)

(ee) y

?

4 x8

x7

ft (?, ‗ ?).

(=) Rrlva tnft fraf lc tna tref`mia clrdah oy f`y tf`ma`t tl tna gurva xy 4 g? f`h gllrhe`fta fxas

es gl`stf`t.

(>) F gurva es meva` oy tna aquftel`s x 4 ft? & y 4 ft7 . F vfrefoia pfer lc parpa`heguifr ieàs tnrlumn

tna lreme` 'L' daat tna gurva ft R & V . Wnlw tnft tna ilgus lc tna ple`t lc e`tarsagtel` lc tna

tf`ma`ts ft R & V es 8y? 4 7fx - f? .

F`swars ; (1) ‗o?

f

(3) (e) Xf`ma`t ; y 4 ?x + 6 , @lrd fi ;x + ?y 4 =

(ee) Xf`ma`t ; ?x + y 4 ?, @lrdfi ;x ‗ ?y 4 3



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Xf`ma`t crld f` ax tar`fi ple`t

Meva` f ple`t R(f, o) wnegn hlas `lt iea l` tna gurva y 4 c(x), tna` tna aquftel` lc plsseoia tf`ma`ts tl

tna gurva y 4 c(x), pfsse`m tnrlumn (f, o) gf` oa clu`h oy slive`m clr tna ple`t lc gl`tfgt V.

c (n) 4fn

o)n(c

F`h aquftel` lc tf`ma`t es y ‗ o 4fn

o)n(c

(x ‗ f)

Axfdpia # 62 Ce`h vfiua lc g sugn tnft ieà kleè`m ple`ts (2, 7) f`h (1, ‗ ?) oagldas tf`ma`t tl gurva

y 46x

g

.

Wliutel` ; Aquftel` lc ieà kle e`m F & O es x + y 4 7

Wlive`m tnes ieà f`h gurva wa mat

7 ‗ x 46x

g

x? ‗ ?x + (g ‗ 7) 4 2 ......( e)

Clr tf`ma`gy, rllts lc tnes aquftel` dust oa aqufi.

Na`ga hesgrede`f`t lc qufhrfteg aquftel` 4 2

8 4 8 (g ‗ 7) g 4 8Rutte`m g 4 8, aquftel` (e) oagldas

x? ‗ ?x + 6 4 2 x 4 6

Na`ga ple`t lc gl`tfgt oagldas (6, ?).

@lta ; Ec f ieà tlugnas f gurva tna` l` slive`m tna aquftel` lc ieà f`h gurva wa mat ft iafst twl rapaftah rllts

glrraspl`he`m tl ple`t lc gl`tfgt.

Axfdpia # 66 Xf`ma`t ft R(?, >) l` tna gurva y 4 x 7 daats tna gurva fmfe` ft V. Ce`h gllrhe`ftas lc V.

Wliutel` ; Aquftel` lc tf`ma`t ft (?, >) es

y 4 6?x ‗ 63

Wlive`m tnes wetn y 4 x 7

x7 ‗ 6?x + 63 4 2

Xnes guoeg weii meva fii ple`ts lc e`tarsagtel` lc ieà f`h gurva y 4 x 7 e.a., ple`t R f`h V. (saa

cemura)

Out, se`ga ieà es tf`ma`t ft R sl x 4 ? weii oa f rapaftah rllt lc aquftel` x7 ‗ 6?x + 63 4 2 f`h

f`ltnar rllt weii oa x 4 n. Qse`m tnalry lc aquftel`s ;

sud lc rllts ? + ? + n 4 2 n 4 ‗ 8

Na`ga gllrhe`ftas lc V fra (‗ 8, ‗ 38)



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"df`esnjudfrpnysegs.e`" =


(5) Nlw df`y tf`ma`ts fra plsseoia crld lreme tl tna gurva y 4 (x + 6)7. Fisl ce`h tna aquftel` lc

tnasa tf`ma`ts.

(62) Ce`h tna aquftel` lc tf`ma`t tl tna nyparolif y 41x

5x

wnegn pfssas tnrlumn (2, 2) lreme`

F`swars ; (5) y 4 2, 8y 4 ?=x (62) x + y 4 2: ?1y + x4 2

Ia`mtns lc tf`ma`t, `lrdfi, suotf`ma`t f`h suo`lrdfi ;

Iat R (n, j) oa f`y ple`t l` gurva y 4 c(x). Iat tf`ma`t hrfw` ft ple`t R daats x-fxes ft X & `lrdfi ft

ple`t R daats x-fxes ft @. Xna` tna ia`mtn RX es gfiiah tna ia`mtn lc tf`ma`t f`h R@ es gfiiah ia`mtn lc

`lrdfi. (fs snlw` e` cemura)

Rrlkagtel` lc samda`t RX l` x-fxes, XD, es gfiiah tna suotf`ma`t f`h sedeifriy prlkagtel` lc ie`a samda t

R@ l` x fxes, D@ es gfiiah suo`lrdfi.

Iat d 4)j,n(hx

hy

4 silpa lc tf`ma`t.

Na`ga aquftel` lc tf`ma`t es d (x ‗ n) 4 (y ‗ j).

Rutte`m y 4 2, wa mat x - e`targapt lc tf`ma`t es x 4 n ‗d

j

Wedeifriy, tna x-e`targapt lc `lrdfi es x 4 n + jd

@lw, ia`mtn RX, R@,XD, D@ gf` oa afseiy avfiuftah use`m hestf`ga clrduif

(e) RX 4?d

66|j| 4 Ia`mtn lc Xf`ma`t

(ee) R@ 4 ?d6|j| 4 Ia`mtn lc @lrdfi

(eee) XD 4d

j4 Ia`mtn lc suotf`ma`t

(ev) D@ 4 |jd| 4 Ia`mtn lc suo`lrdfi.

Axfdpia # 6? Ce`h tna ia`mtn lc tf`ma`t clr tna gurva y 4 x 7 + 7x? + 8x ‗ 6 ft ple`t x 4 2.

Wliutel` ; Nara, d 42xhx

hy

hx

hy4 7x? + 3x + 8 d 4 8



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"dfèsnjudfrpnysegs.e`" >

f`h, j 4 y(2) j 4 ‗ 6

4 ?d

66|j| 4

63

66|)6(| 4

8

6=

Axfdpia # 67 Rrlva tnft clr tna gurva y 4 oa x/f, tna ia`mtn lc suotf`ma`t ft f`y ple`t es fiwfys gl`stf`t.

Wliutel` ; y 4 oax/f

Iat tna ple`t oa (x6

, y6

)

d 46xxhx

hy

4

f

a.o f/x6

f

y6

@lw, ia`mtn lc suotf`ma`t 4d

y64

f/y

y

6

64 | f | : wnegn es fiwfys gl`stf`t.

Axfdpia # 68 Clr tna gurva y 4 f ` (x? ‗ f?) snlw tnft sud lc ia`mtns lc tf`ma`t & suotf`ma`t ft f`y ple`t

es prlplrtel`fi tl gllrhe`ftas lc ple`t lc tf`ma`gy.

Wliutel` ; Iat ple`t lc tf`ma`gy oa (x6, y

6)

d 46xxhx

hy

4 ?

6?

6

fx

fx?

Ia`mtn lc tf`ma`t + suotf`ma`t 4 |y6| ?d

66 +

d

y6

4 |y6| ?

6?

??6

?

xf8

)fx(6

+

6

??66

fx?

)fx(y 4 |y

6|

|fx|?

xf?fx

6

?6

?886

+6

??66

fx?

)fx(y

4 6

??66

fx?

)fx(y

+ 6

??66

fx?

)fx(y

4 |fx|?

)x?(|y|

6

?66

4 f

yx 66


(66) Clr tna gurva xd + ` 4 fd ‗ ` y?`, wnara f es f plseteva gl`stf`t f`h d, ` fra plseteva e`tamars,

prlva tnft tna dtn plwar lc suotf`ma`t vfreas fs `tn plwar lc suo`lrdfi.

(6?) Rrlva tnft tna samda`t lc tna tf`ma`t tl tna gurva y 4?

f`

??

??

xff

xff

‗ ?? xf gl`tfeàh

oatwaa` tna y-fxes & tna ple`t lc tf`ma`gy nfs f gl`stf`t ia`mtn .

(67) Ce`h tna ia`mtn lc tna suo`lrdfi tl tna gurva y? 4 x7 ft tna ple`t (8, >).

F`swar ; (67) ?8

F`mia oatwaa` tna gurvas

F`mia oatwaa` twl e`tarsagte`m gurvas es haceàh fs tna fguta f`mia oatwaa` tnaer tf`ma`ts (lr `lrdfis) ft

tna ple t lc e`tarsagtel` lc twl gurvas (fs snlw` e` cemura).

tf` 4?6

?6

dd6

dd

wnara d6 & d

? fra tna silpas lc tf`ma`ts ft tna e`tarsagtel` ple`t (x

6, y

6).



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@ltas ; (e) Xna f`mia es hace ah oatwaa` twl gurvas ec tna gurvas fra e tarsagte`m. Xnes gf` oa a`surah oy

ce`he`m tnaer ple t lc e`tarsagtel` lr mrfpnegfiiy.

(ee) Ec tna gurvase`tarsagt ft dlra tnf` là ple t tna` f`mia oatwaa` gurvas es clu`h lut wetn raspagt tl

tna ple`t lc e`tarsagtel`.

(eee) Xwl gurvas fra sfeh tl oa lrtnlml`fi ec f`mia oatwaa` tnad ft afgn ple`t lc e`tarsagtel` es remnt

f`mia. e.a. d6 d

?4 ‗ 6.

Axfdpia # 61 Ce`h f`mia oatwaa` y? 4 8x f`h x? 4 8y. Fra tnasa twl gurvas lrtnlml`fi<

Wliutel` ; y? 4 8x f`h x? 4 8y e`tarsagt ft ple`t (2, 2) f`h (8, 8) (saa cemura).

G6

; y? 4 8x G?

; x? 4 8y

hx

hy4

y

?

hx

hy4

?

x

)2,2(hx

hy

)2,2(hx

hy4 2

Na`ga tf` 4 52² ft ple`t (2, 2)

)8,8(hx

hy4

?

6

)8,8(hx

hy4 ?

tf` 4

?

6.?6

?

6?

4

8

7

Xwl gurvas fra `lt lrtnlml`fi oagfusa f`mia oatwaa` tnad ft (8, 8) es lt 52².

Axfdpia # 63 Ce`h tna f`mia oatwaa` gurvas y? 4 8x f`h y 4 a ‗x/?

Wliutel` ; Iat tna gurvas e`tarsagt ft ple`t (x6, y

6) (saa cemura).

clr y? 4 8x,)y,x( 66

hx

hy4

6y

?

f`hclr y 4 a ‗x/?

)y,x( 66hx

hy4 ‗

?

6?/x6a 4 ‗

?

y6

d6 d? 4 ‗ 6 Na`ga 4 52²

@lta tnft wa nfva `lt fgtufiiy clu`h tna e`tarsagtel` ple`t out maldatregfiiy wa gf` saa tnft tna gurvas e`tarsagt.

Axfdpia # 6= Ce`h plsseoia vfiuas lc p sugn tnft tna aquftel` px? 4 `x nfs axfgtiy là sliutel`.

Wliutel` ; Xwl gurvas dust e`tarsagt ft lìy là ple`t.

(e) (ee)



DFXNW


E. Ec p 2 tna` tnara axests lìy là sliutel` (saa mrfpn - (e))

EE. Ec p 0 2

tna` tna twl gurvas dust lìy tlugn afgn ltnar

e.a. tf`ma`t ft y 4 px? f`h y 4 `x dust nfva sfda silpa ft ple`t (x6, y

6)

?px6 4

6x

6

x6

? 4p?

6 .........(e)

fisl y6 4 px

6? y

6 4 p

p?

6

y6 4

?

6........(ee)

f`h y6 4 `x

6

?

64 `x

6

x6 4 a6/? .........(eee)

x6

? 4p?

6 a 4

p?

6 p 4

a?

6

Na`ga plsseoia vfiuas lc p fra (‗ , 2T

a?

6

Waic prfgtega prloiad ;

(68) Ce`h tna f`mia lc e`tarsagtel` lc tna cliilwe`m gurvas;

(e) y 4 x? & 3y 4 = x7 ft (6, 6)

(ee) x? ‗ y? 4 1 &>

y

6>

x ??

4 6.

F`swars ; (68) (e) /? (ee) /?

Wnlrtast hestf`ga oatwaa` twl gurvas

Wnlrtast hestf`ga oatwaa` twl `l -e tarsagte m heccara tefoia gurvas es fiwfys fil`m tnaer glddl` `lrdfi.

(]naravar haceàh)

Axfdpia # 6> Ce`h tna snlrtast hestf`ga oatwaa` tna ieà y 4 x ‗ ? f`h tna pfrfolif y 4 x? + 7x + ?.

Wliutel` ; Iat R(x6, y

6) oa f ple`t gilsast tl tna ieà y 4 x ‗ ?

tna`)y,x( 66

hx

hy4 silpa lc ieà

?x6 + 7 4 6 x

6 4 ‗ 6 y

6 4 2

Na`ga ple t (‗ 6, 2) es tna gilsast f`h ets parpa`heguifr hestf`ga crld tna ie a y 4 x ‗ ? weii meva tna

snlrtast hestf`ga



DFXNW


p 4?

7.

Waic prfgtega prloiad ;

(61) Ce`h tna deèdud & dfxedud vfiuas lc (x + ?)? + (y ‗ 6)?, ec (x ‗ ?)? + (y + 6)? 8.

F`swar ; (61) ?1? , ?1? .

Arrlr f`h Fpprlx edftel` ;

Iat y 4 c(x) oa f cu`gtel`. Ec tnasa es f` arrlr x e` x tna` glrraspl`he`m arrlr e` y es y 4 c(x + x) ‗ c(x).

]a nfva2x

ied x

)x(c )xx(c

4hx

hy4 c (x)

]a haceà tna heccara`tefi lc y, ft ple`t x, glrraspl`he`m tl tna e`grada`t x fs c (x) x f`h ha`lta et oy hy.

e.a. hy 4 c (x) x.

Iat R(x, c(x)), V((x + x), c(x + x)) (fs snlw` e` cemura)

y 4 VW,

x 4 RW,

hy 4 _W

E` df`y prfgtegfi setuftel`s, et es afsear tl avfiufta hy out `lt y.

_liia―s Xnalrad ;

Ec f cu`gtel` c haceàh l` Zf, oT es

(e) gl`teùlus l` Zf, oT

(ee) harevfoia l` (f, o) f`h

(eee) c(f) 4 c(o),

tna` tnara axests ft iafst là rafi ùdoar g oatwaa` f f`h o (f 9 g 9 o) sugn tnft c (g) 4 2

Maldatregfi Axpif`ftel` lc _liia ―s Xnalrad ;

Iat tna gurva y 4 c(x), wnegn es gl`teùlus l` Zf, oT f`h harevfoia l` (f, o), oa hrfw` (fs s nlw` e` cemura).

F(f, c(f)), O(o, c(o)), c(f) 4 c(o), G(g, c(g)), c (g) 4 2.

G6 (g

6, c(g

6)), c (g

6) 4 2

G? (g

?, c(g

?)), c (g

?) 4 2

G7 (g

7, c(g

7)), c (g

7) 4 2

Xna tnalrad sedpiy stftas tnft oatwaa` twl ple ts wetn aqufi lrhe ftas l` tna mrfpn lc c(x), tnara axests ft

iafst là ple`t wnara tna tf`ma`t es pfrfiiai tl x-fxes.



DFXNW

"dfèsnjudfrpnysegs.e`" 6?

Fimaorfeg E`tarpratftel` lc _liia―s Xnalrad ;

Oatwaa` twl zarls f f`h o lc c(x) (e.a. oatwaa` twl rllts f f`h o lc c(x) 4 2) tnara axests ftiafstlà zarl lcc (x)

Axfdpia # 65 ; Sarecy _liia―s tnalrad clr c(x) 4 (x ‗ f)` (x ‗ o)d , wnara d, ` fra plseteva rafi ùdoars, clr

x Zf, oT.

Wliutel` ; Oae`m f pliy ldefi cu`gtel` c(x) es gl`teùlus fs waii fs heccara`tefoia. Fisl c(f) 4 c(o)

c (x) 4 2 clr slda x (f , o)

`(x ‗ f)`‗6 (x ‗ o)d + d(x ‗ f)` (x ‗ o)d‗6 4 2

(x ‗ f)`‗6 (x ‗ o)d‗6 Z(d + `) x ‗ (ò + df)T 4 2

x 4`d

dfò

, wnegn ieas e` tna e`tarvfi (f, o), fs d, ` _+.

Axfdpia # ?2 ; Ec ?f + 7o + 3g 4 2 tna` prlva tnft tna aquftel` fx ? + ox + g 4 2 nfs ft iafst là rafi rllt

oatwaa` 2 f`h 6.

Wliutel` ; Iat c(x) 4

7

fx7

+

?

ox?

+ gx

c(2) 4 2

f`h c(6) 47

f+

?

o+ g 4 ?f + 7o + 3g 4 2

Ec c(2) 4 c(6) tna` c (x) 4 2 clr slda vfiua lc x (2, 6)

fx? + ox + g 4 2 clr ft iafst là x (2, 6)

Waic Rrfgtega Rrloiads ;

(63) Ec c(x) sftesceas gl`hetel` e` _liia―s tnalrad tna` snlw tnft oatwaa` twl gl`saguteva zarls lc c (x)

tnara ieas ft dlst là zarl lc c(x).

(6=) Wnlw tnft clr f`y rafi ùdoars , tna pliy`ldefi R(x) 4 x= + x7 + , nfs axfgtiy là rafi rllt.

Ifmrf`ma―s Daf` Sfiua Xnalrad IDSX ;

Ec f cu`gtel` c haceàh l` Zf, oT es

(e) gl`teùlus l` Zf, oT f`h

(ee) harevfoia l` (f, o)

tna` tnara axests ft iafst là rafi ùdoars oatwaa` f f`h o (f 9 g 9 o) sugn tnftfo

)f(c )o(c

4 c (g)

Rrllc ; Iat us gl`sehar f cu`gtel` m(x) 4 c(x) + x, x Zf, oTwnara es f gl`stf`t tl o hatardeàh sugn tnft m(f) 4 m(o).

4 ‗fo

)f(c )o(c

@lw tna cu`gtel` m(x), oae m tna sud lc twl gl teùlus f`h harevfoia cu`gtel`s et saic

(e) gl`teùlus l` Zf, oT

(ee) harevfoia l` (f, o) f`h

(eee) m(f) 4 m(o).

Xnaraclra, oy _liia―s tnalrad tnara axests f rafi ùdoar g (f, o) sugn tnft m(g) 4 2

Out m(x) 4 c (x) + 2 4 m(g) 4 c (g) +

c (g) 4 ‗ 4fo

)f(c )o(c



DFXNW


Maldatregfi E`tarpratftel` lc IDSX ;

Xna tnalrad sedpiy stftas tnft oatwaa` twl ple tsF f`h O lc tna mrfpn lc c(x) tnara axests ft iafst là ple t

wnara tf`ma`t es pfrfiiai tl gnlrh FO.

G(g, c(g)), c (g) 4 silpa lc FO.

Fitar`fteva Wtftada`t ; Ec e` tna stftada`t lc IDSX, o es rapifgah oy f + n, tna` ùdoar g oatwaa` f f`h o

dfy oa wretta` fs f + n, wnara 2 9 9 6. Xnus

n

)f(c )nf(c 4 c (f + n)

lr

c(f + n) 4 c(f) + nc (f + n), 2 9 9 6

Axfdpia # ?6 ; Sarecy IDSX clr c(x) 4 ‗ x? + 8x ‗ 1 f`h x Z‗6, 6T

Wliutel` ; c(6) 4 ‗? : c(‗6) 4 ‗62

c (g) 4)6(6

)6(c )6(c

‗?g + 8 4 8 g 4 2

Axfdpia # ?? ; Qse`m Ifmrf`ma―s daf` vfiua tnalrad, prlva tnft ec o 0 f 0 2,

tna` ?o6

fo

9 tf` ‗6 o ‗ tf` ‗6 f 9 ?f6

fo

Wliutel` ; Iat c(x) 4 tf` ‗6 x : x Zf, oT fppiye`m IDSX

c (g) 4fo

ftfòtf` 66

clr f 9 g 9 o f`h c (x) 4 ?x6

,

@lw c (x) es f dl`ltlègfiiy hagrafse m cu`gtel`

Na ga ec f 9 g 9 o c (o) 9 c (g) 9 c (f)

?o6

6

9

fo

ftfòtf` 66

9 ?f6

6

Na`ga prlvah

Axfdpia # ?7 ; Iat c ;_ _ oa f twega heccara tefoia cu`gtel` sugn tnft c(?) 4 >, c(8) 0 38, c(=) 4 787 tna` snlw tnft

tnara axests f g (?, =) sugn tnft c (g) 9 3g.

Wliutel`; Gl`sehar m(x) 4 c(x) ‗ x7

Oy IDSX

?8

)?(m)8(m

4 m(g6) , ? 9 g

6 9 8

f`h

8=

)8(m)=(m

4 m(g?), 8 9 g

? 9 =

m(g6) 0 2 , m(g

?) 9 2

Oy IDSX



DFXNW


6?

6?

gg

)g(m)g(m

4 m(g), g

6 9 g 9 g

?

m(g) 9 2

c (g) ‗ 3g 9 2 clr sfda g (g6

, g?) g (?, =)

Waic Rrfgtega Rrloiads

(6>) Ec f cu`gtel` c(x) sftesceas tna gl`hetel`s lc IDSX f`h c (x) 4 2 clr fii x (f, o), tna` c(x) es gl`stf`t

l` Zf, oT.

(65) Qse`m IDSX, prlva tnft ec twl cu`gtel`s nfva aqufi harevftevas ft fii ple ts lc (f, o), tna` tnay heccar

oy f gl`stf`t

(?2) Ec f cu`gtel` c es

(e) gl`te ulus l` Zf, oT,

(ee) harevfoia l` (f, o) f`h (eee) c (x) 0 2, x (f, o), tna` snlw tnft c(x) es stregtiy e`grafse`m l` Zf, oT.

Dl`ltlègety lc f cu`gtel` ;

Iat c oa f rafi vfiuah cu`gtel` nfve`m hldfe H(H _) f`h W oa f suosat lc H. c es sfeh tl oa dl`ltlègfiiy

e`grafse`m (`l` hagrafse`m) (e`grafse`m) e` W ec clr avary x6, x

? W, x

6 9 x

? c( x

6) c( x

?). c es sfeh tl oa

dl`ltlègfiiy hagrafse`m (`l` e grafse`m) (hagrafse`m) e` W ec clr avary x6, x

? W, x

6 9 x

? c(x

6) c(x

?)

c es sfeh tl oa stregtiy e`grafse`m e` W ec clr x6, x

? W, x

6 9 x

? c(x

6) 9 c(x

?). Wedeifriy, c es sfeh tl oa

stregtiy hagrafse`m e` W ec clr x6, x

? W, x

6 9 x

? c(x

6) 0 c(x

?) .

@ltas ; ( e) c es stregtiy e`grafse m c es dl`ltlègfiiy e`grafse m (`l` hagrafse`m). Out gl`varsa àah

`lt oa trua.

(ee) c es stregtiy hagrafse m c es d l`ltlègfiiy hagrafse`m (`l` e`grafse`m). Fmfe`, gl`varsa àah

`lt oa trua.

(eee) Ec c(x) 4 gl`stf`t e` W, tna` c es e grafse`m fs waii fs hagrafse`m e` W

(ev) F cu`gtel` c es sfeh tl oa f` e grafse`m cu`gtel` ec et es e grafse`m e` tna hldfe`. Wedeifriy, ec c es

hagrafse`m e` tna hldfe`, wa sfy tnft c es dl`ltlègfiiy hagrafse`m

(v) c es sfeh tl oa f dl`ltl eg cu`gtel` ec aetnar et es dl`ltlègfiiy e grafse`m lr dl`ltlègfiiy

hagrafse`m

(ve) Ec c es e grafse`m e f suosat lc W f`h hagrafse m e f`ltnar suosat lc W, tna` c es `l` dl`ltlèg

e` W.

Fppi egft el` lc h ec ca ra`t ef tel` clr hatagt e`m dl`ltl` eg et y ;

Iat oa f` e`tarvfi (lpa` lr gilsah lr sade lpa` f`h sade gilsah)

(e) Ec c (x) 0 2 x , tna` c es stregtiy e`grafse`m e` (ee) Ec c (x) 9 2 x , tna` c es stregtiy hagrafse`m e`

@lta ; Iat E oa f` e`tarvfi (lr rfy) wnegn es f suosat lc hldfe` lc c. Ec c (x) 0 2, x axgapt clr glu`tfoiy

df`y ple`ts wnara c (x) 4 2, tna` c(x) es stregtiy e`grafse`m e` .{c (x) 4 2 ft glu`tfoiy df`y ple`ts c (x) 4 2 hlas `lt lggur l` f` e`tarvfi wnegn es f suosat lc }



DFXNW


Axfdpia # ?8 ; Iat c(x) 4 x7. Ce`h tna e`tarvfis lc dl`ltlègety.

Wliutel` ; c (x) 4 7x?

c (x) 0 2 avarywnara axgapt ft x 4 2. Na`ga c(x) weii oa stregtiy e`grafse`m cu`gtel` clr

x _ {saa cemura}

Axfdpia # ?1 ; Iat c(x) 4 x ‗ se x. Ce`h tna e`tarvfis lc dl`ltl egety.

Wliutel` ; c (x) 4 6 ‗ glsx

@lw, c (x) 0 2 avary wnara, axgapt ft x 4 2, ¶ ?, ¶ 8 atg. Out fii tnasa ple`ts fra hesgrata

(glu`tfoia) f`h hl `lt clrd f` e`tarvfi. Na`ga wa gf` gl`giuha tnft c(x) es stregtiy e`grafse`m

e` _. E` cfgt wa gf` fisl saa et mrfpnegfiiy.

Axfdpia # ?3 ; Iat us gl`sehar f`ltnar cu`gtel` wnlsa mrfpn es snlw` oailw clr x (f, o).

Wliutel` ; Nara fisl c (x) 2 clr fii x (f, o). Out, `lta tnft e` tnes gfsa, c (x) 4 2 nlihs clr fii x (g, h)

f`h (a, o). Xnus tna meva` cu`gtel` es e`grafse`m (dl`ltlègfiiy e`grafse`m) e` (f, o), out `lt

stregtiy e`grafse`m.

Axfdpia # ?= ; Ce`h tna e`tarvfis e` wnegn c(x) 4 x7 ‗ 7x + ? es e`gr afse`m.

Wliutel` ; c(x) 4 x7 ‗ 7x + ?

c (x) 4 7(x? ‗ 6)

c (x) 4 7(x ‗ 6) (x + 6)

clr D.. c (x) 2 7(x ‗ 6) (x + 6) 2

x ( ‗ , ‗ 6T Z6, ), tnus c es e`grafse`m e` (‗ , ‗6T f`h fisl e` Z6, )

Axfdpia # ?> ; Ce`h tna e`tarvfis lc dl`ltlègety lc tna cliilwe`m cu`gtel`s.

(e) c(x) 4 x?

(x ‗ ?)?

(ee) c(x) 4 x

` x(eee) c(x) 4 se`x + glsx : x Z2, ?T

Wliutel` ; (e) c(x) 4 x? (x ‗ ?)?



DFXNW


c (x) 4 8x (x ‗ 6) (x ‗ ?)

losarve`m tna sem` gnf`ma lc c (x)

Na`ga e`grafse`m e Z2, 6T f`h e` Z?, )

f`h hag rafse`m clr x (‗ , 2T f`h Z6, ?T

(ee) c(x) 4 x ` x

c (x) 4 6 + ` x

c (x) 2 ` x ‗ 6 x a

6

e`grafse`m clr x

,

a

6f`h hagrafse`m clr x

a

6,2 .

( eee) c (x ) 4 s e x + g ls x

c (x) 4 glsx ‗ se`x

clr e`grafse`m c (x) 2 glsx se`x

c es e`grafse`m e

8

,2 f`h

?,

8

1

c es hagrafse`m e`

8

1,

8

@lta ; Ec f cu`gtel` c(x) es e`grafse`m e` (f, o) f`h c(x) es gl`teùlus e` Zf, oT, tna` c(x) es e`grafse`m l` Zf, oT

Axfdpia # ?5; c(x) 4 ZxT es f stap up cu`gtel`. Es et f stregtiy e`grafse`m cu`gtel` clr x _.

Wliutel` ; @l, c(x) 4 ZxT es e`grafse m (dl`ltl egfiiy e`grafse m) (`l -hagrafse m), out lt stregtiy e`grafse m

cu`gtel` fs eiiustrftah oy ets mrfpn.

Axfdpia # 72 ; Ec c(x) 4 se`8x + gls8x + ox + g, tna` ce`h plsseoia vfiuas lc o f`h g sugn tnft c(x) es dl`ltlèg

clr fii x _

Wliutel` ; c(x) 4 se`8x + gls8x + ox + g

c (x) 4 8 se`7x glsx ‗ 8gls7x se`x + o 4 ‗ se`8x + o.

Gfsa - ( e) ; clr D.E. c (x) 2 clr fii x _

o se`8x clr fii x _ o 6

Gfsa - (ee) ; clr D.H. c (x) 2 clr fii x _

o se`8x clr fii x _ o ‗ 6

Na`ga clr c(x) tl oa d l`ltlèg o (‗ , ‗ 6T Z6, ) f`h g _.

Axfdpia # 76;Ce`h plsseoia vfiuas lc 'f' sugn tnft c(x) 4 a ?x ‗ (f + 6) ax + ?x es dl`ltlègfiiy e`grafse`m clr

x _

Wliutel` ; c(x) 4 a?x ‗ (f + 6) ax + ?x

c (x) 4 ?a?x ‗ (f + 6) ax + ?

@lw, ?a?x ‗ (f + 6) ax + ? 2 clr fii x _



DFXNW

"dfèsnjudfrpnysegs.e`" 6=

?

x

x

a

6a ‗ (f + 6) 2 clr fii x _

(f + 6) 9 ?

x

x

a

6a clr fi i x _

f + 6 8

?vfiuaedudde`nfs

a

6a

x

x

f 7

F i e t ar (Qse m mrfpn)

?a?x ‗ (f + 6) ax + ? 2 clr fii x _

putte`m ax 4 t : t (2, )

?t? ‗ (f + 6) t + ? 2 clr fii t (2, )

Gfsa - (e) ; H 2

(f + 6)? ‗ 8 2

(f + 1) (f ‗ 7) 2

f Z‗ 1, 7T

lr

Gfsa - (ee) ; oltn rllts fra l` plseteva

H 2 & ‗f?

o9 2 & c(2) 2

f (‗ , ‗ 1T Z7, ) &8

6f 9 2 & ? 2

f (‗ , ‗ 1T Z7, ) & f 9 ‗ 6 & f _

f (‗ , ‗ 1T

Xfje`m uèl` lc (e) f`h (ee), wa mat f (‗ , 7T.


(?6) Ce`h tna e`tarvfis lc dl`ltlègety lc tna cliilwe`m cu`gtel`s.

(e) c(x) 4 ‗ x7 + 3x? ‗ 5x ‗ ? (ee) c(x ) 4 x +6x

6

(eee) c(x) 4 x . ?xxa (ev) c (x) 4 x ‗ glsx

(??) Iat c(x) 4 x ‗ tf` ‗6x. Rrlva tnft c(x) es dl`ltlègfiiy e`grafse`m clr x _.

(?7) Ec c(x) 4 ?ax ‗ fa ‗x + (?f + 6) x ‗ 7 dl`ltlègfiiy e`grafsas clr x _, tna` ce`h rf`ma lc vfiuas

lc f

(?8) Iat c (x ) 4 a?x ‗ fax + 6. Rrlva tnft c(x) gf``lt oa dl`ltlègfiiy hagrafse`m clr x _ clr f`yvfiua lc 'f'.

(?1) Xna vfiuas lc 'f' clr wnegn cu`gtel` c(x) 4 (f + ?) x7 ‗ fx ? + 5fx ‗ 6 dl`ltlègfiiy hagrafse m clr

x _.

F`swars ; (?6) (e) e` Z6, 7T : H e` (‗ , 6T (7, )

(ee) e` (‗ , ‗ ?T Z2, ) :H e` Z‗ ?, ‗ 6) (‗6, 2T

(eee) e`

6,

?

6: H e`

?

6, Z6, )

(ev) E clr x _

(?7) f 2 (?1) ‗ 9 f ‗ 7



DFXNW

"dfèsnjudfrpnysegs.e`" 6>

Dl`ltlègety lc cu`gtel` folut f ple`t ;

6. F cu`gtel` c(x) es gfiiah fs f stregtiy e`grafse`m cu`gtel` folut f ple`t (lr ft f ple`t) f Hc ec et es stregtiy

e`grafse`m e` f` lpa` e`tarvfi gl`tfeè`m f (fs snlw` e` cemura).

?. F cu`gtel` c(x) es gfiiah f stregtiy hagrafse`m cu`gtel` folut f ple`t x 4 f, ec et es stregtiy hagrafse`m e

f` lpa` e`tarvfi gl`tfeè`m f (fs snlw` e` cemura).

@lta ; Ec x 4 f es f olu`hfry ple`t tna` usa tna fpprlprefta là sehah eàqufiety tl tast dl`ltlègety lc c(x).

a.m. ; ]negn lc tna cliilwe`m cu`gtel`s (fs snlw` e` cemura) es e`grafse`m, hagrafse`m lr àetnar e`grafse`m `lr

hagrafse`m ft

x 4 f.

(e) (ee)



DFXNW


(eee) (ev)

Xast clr e`g rafse`m f`h hag rafs e`m cu`gtel`s folut f ple`t

Iat c(x) oa heccara`tefoia.

(6) Ec c (f) 0 2 tna` c(x) es e`grafse`m ft x 4 f.

(?) Ec c (f) 9 2 tna` c(x) es hagrafse`m ft x 4 f.

(7) Ec c (f) 4 2 tna` axfdeà tna sem` lc c (x) l` tna iact àemnolurnllh f`h tna remnt àemnolurnllh

lc f.

(e) Ec c (x) es plseteva l` oltn tna àemnolurnllhs, tna` c es e`grafse`m ft x 4 f.

(ee) Ec c (x) es àmfteva l` oltn tna àemnolurnllhs, tna` c es hagrafse`m ft x 4 f.

(eee) Ec c (x) nfva lpplseta sem`s l` tnasa àemnolurnllhs, tna` c es `l`-dl`ltlèg ft x 4 f.

Axfdpia # 7?; Iat c(x) 4 x7 ‗ 7x + ?. Axfdeà tna dl`ltlègety lc cu`g tel` ft ple`ts x 4 2, 6, ?.

Wliutel` ; c(x) 4 x7 ‗ 7x + ?

c (x) 4 7(x? ‗ 6)

(e) c (2) 4 ‗ 7 hagrafse`m ft x 4 2

(ee) c (6) 4 2

fisl, c (x) es plseteva l` iact àemnolurnllh f`h c (x) es àmfteva e` remnt àemnolurnllh.

àetnar e`grafse`m `lr hagrafse`m ft x 4 6.

(eee) c (?) 4 5 e`grafse`m ft x 4 ?

@lta ; Folva datnlh es fppiegfoia lìy clr cu`gtel`s tnlsa fra gl`teùlus ft x 4 f.


(?3) Clr afgn lc tna cliilwe`m mrfpn gldda`t l` dl`ltlègety lc c(x) ft x 4 f.

(e) (ee) (eee) (ev)

(?=) Iat c (x ) 4 x7 ‗ 7x? + 7x + 8, gldda`t l` tna dl`ltlèg oanfvelur lc c(x) ft (e) x 4 2 (ee) x 4 6.

(?>) Hrfw tna mrfpn lc cu`gtel` c(x) 4

?x6TxZ

6x2x. Mrfpnegfiiy gldda`t l` tna dl`ltlèg

oanfvelur lc c(x) ft x 4 2, 6, ?. Es c(x) D. . clr x Z2, ?T <

F`swars ; (?3) (e) àetnar D.. lr D.H. (ee) D.H. (eee) D.H (ev) D..

(?=) D.. oltn ft x 4 2 f`h x 4 6.

(?>) D.. ft x 4 2, ? : àetnar D.. `lr D.H. ft x 4 6. @l, c(x) es `lt D.. clr x Z2, ?T.



DFXNW

"dfèsnjudfrpnysegs.e`" ?2

Qsa lc dl`ltlège ty clr p rlve`m eàquf ie teas

Gldpfresl` lc twl cu`gtel`s c(x) f`h m(x) gf` oa hlà oy f`fiyse`m tna dl`ltlèg oanfvelur lc

n(x) 4 c(x) ‗ m(x)

Axfdpia # 77 ; Clr x

?,2 prlva tnft se` x 9 x 9 tf` x

Wliutel` ; Iat c(x) 4 x ‗ se` x c (x) 4 6 ‗ gls x

c (x) 0 2 clr x

?

,2

c(x) es D.E. c(x) 0 c(2)

x ‗ se` x 0 2 x 0 se` x

Wedeifriy gl`sehar f`ltnar cu`gtel` m(x) 4 x ‗ tf` x m(x) 4 6 ‗ sag?x

m(x) 9 2 clr x

?,2

m(x) es D.H.

Na`ga m(x) 9 m(2)

x ‗ tf` x 9 2 x 9 tf` x

se` x 9 x 9 tf` x Na`ga prlvah

Axfdpia # 78 ; Clr x (2, 6) prlva tnft x ‗7

x7

9 tf` ‗6 x 9 x ‗3

x7

na`ga lr ltnarwesa ce`h2x

ied

x

xtf` 6

Wliutel` ; Iat c(x) 4 x ‗7

x7

‗ tf` ‗6x

c(x) 4 6 ‗ x? ‗ ?x6

6

c (x) 4 ‗?

8

x6

x

c (x) 9 2 clr x (2, 6) c(x) es D.H.

c(x) 9 c(2)

x ‗7

x7

‗ tf` ‗6x 9 2

x ‗7

x7

9 tf` ‗6x ...........(e)

Wedeifriy m(x) 4 x ‗3

x7

‗ tf` ‗6x

m(x) 4 6 ‗?

x?

‗ ?x6

6

m(x) 4)x6(?

)x6(x?

??

m(x) 0 2 clr x (2, 6) m(x) es D.E. m(x) 0 m(2)



DFXNW


x ‗3

x7

‗ tf` ‗6x 0 2

x ‗3

x7

0 tf` ‗6x ........(ee)

crld (e) f`h (ee), wa mat

x ‗7

x7

9 tf` ‗6x 9 x ‗3

x7

Na`ga Rrlvah

Fisl, 6 ‗7

x?

9x

xtf` 6

9 6 ‗3

x ?

, clr x 0 2

Na`ga oy sf`hwegn tnalrad wa gf` prlva tnft2x

ied x

xtf` 6

4 6 out et dust fisl oa `ltah tnft

fs x 2, vfiua lc x

xtf` 6

6 crld iact nf`h seha e.a.x

xtf` 6

9 6

2x

ied

x

xtf` 6

4 2

@LXA ; E` prlve`m eàqufieteas, wa dust fiwfys gnagj wna` hlas tna aqufiety tfjas pifga oagfusa tna ple`t lc

aqufiety es vary edplrtf t e` tnes d atnlh. @lrdfiiy ple`t lc aqufiety lggur ft a`h ple`t lc tna e`tarvfi lr

weii oa afseiy prahegtah oy net f`h trefi.

Axfdpia # 71 ; Clr x

?

,2 , prlva tnft se` x 0 x ‗3

x7

Wliutel` ; Iat c(x) 4 se` x ‗ x +3

x7

c (x) 4 gls x ‗ 6 +?

x ?

wa gf``lt hageha ft tnes ple`t wnatnar c (x) es plseteva lr àmfteva, na`ga iat us gnagj clr

dl`ltlèg `ftura lc c (x)

c (x) 4 x ‗ se`x

We`ga c (x) 0 2 c (x) es D.E. clr x

?,2

c (x) 0 c (2)

c (x) 0 2 c(x) es D..

c(x) 0 c(2)

se` x ‗ x +3

x7

0 2

se` x 0 x ‗3

x7

Na`ga prlvah



DFXNW

"dfèsnjudfrpnysegs.e`" ??

Axfdpia # 73 ; Axfdeà wnegn es mraftar ; se` x tf` x lr x ?. Na`ga avfiufta2x

ied

?x

xtf`xse`, wnara

x

?

,2

Wliutel` ; Iat c (x ) 4 s e x tf`x ‗ x?

c (x) 4 gls x . tf` x + se` x . sag?x ‗ ?x

c (x) 4 se` x + se` x sag?x ‗ ?x

c (x) 4 gls x + gls x sag ?x + ?sag?x se` x tf` x ‗ ? c (x) 4 (gls x + sag x ‗ ?) + ? sag ?x se` x tf` x

@lw gls x + sag x ‗ ? 4 ?xsagxgls f`h ? sag?x tf` x . se` x 0 2 oagfusa x

?

,2

c (x) 0 2 c (x) es D.E.

Na`ga c (x) 0 c (2)

c (x) 0 2 c(x) es D.E.

c(x) 0 2 se` x tf` x ‗ x? 0 2

Na`ga se` x tf` x 0 x?

?x

xtf`xse`0 6

2xied

?x

xtf`xse`4 6

Axfdpia # 7= ; Rrlva tnft c(x) 4

x

x

66

es dl`ltlègfiiy e`grafse`m e` ets hldfe . Na`ga lr ltnarwesa hrfw

mrfpn lc c(x) f`h ce`h ets rf`ma

Wliutel` ; c(x) 4

x

x

66

, clr Hldfe` lc c(x), 6 +

x

60 2

x

6x 0 2 (‗, ‗6) (2, )

Gl`sehar c (x) 4

x

x

66

?x

6

x

66

x

x

66`

c (x) 4

x

x

66

6x

6

x

66`

@lw

x

x

66

es fiwfys plseteva, na`ga tna sem` lc c (x) hapa`hs l` sem` lc `

x

66 ‗

x6

6

e .a. wa nfva tl gldpfra `

x

66 f`h

x6

6

Wl iats fssuda m(x) 4 `

x

66 ‗

6x

6

m(x) 4

x

66

6

?x

6+ ?)6x(

6

m(x) 4 ?)6x(x

6

(e) clr x (2, ), m(x) 9 2 m(x) es D.H. clr x (2, )

m(x) 0 xied m(x)

m(x) 0 2.



DFXNW


f`h s e ga m(x ) 0 2 c (x) 0 2

(ee) clr x (‗ , ‗ 6), m(x) 0 2 m(x) es D.E. clr x (‗ , ‗6)

m(x) 0 xied m(x)

m(x) 0 2 c (x) 0 2

Na`ga crld (e) f`h (ee) wa mat c (x) 0 2 clr fii x (‗ , ‗6) (2, )

c(x) es D.E. e` ets Hldfe`

Clr hrfwe`m tna mrfpn lc c(x), ets edplrtf`t tl ce`h tna vfiua lc c(x) ft olu`hfry ple`ts

e.a. ¶ , 2, ‗6

xied

x

x

66

4 a

2xied

x

x

66

4 6 f`h

6xied

x

x

66

4

sl tna mrfpn lc c(x) es

_f`ma es y (6, ) ‗ {a}

Axfdpia # 7> ; Gldpfra wnegn lc tna twl es mraftar (622)6/622 lr (626)6/626.

Wliutel` ; Fssuda c(x) 4 x6/x f`h iat us axfdeà dl`ltlèg `ftura lc c(x)

c (x) 4 x6/x .

?x

`x6

c (x) 0 2 x (2,a)

f`h c (x) 9 2 x (a,)

Na`ga c(x) es D.H. clr x a

f`h se`ga 622 9 626

c(622) 0 c(626)

(622)6/622 0 (626)6/626


(?5) Rrlva tna cliilwe`m eàqufieteas

(e) x 9 ‗ `(6 ‗ x) clr x (2, 6)

(ee) x 0 tf` ‗6(x) clr x

( 2,

)

(eee) ax 0 x + 6 clr x ( 2, )

(ev)x6

x

` (6 + x) x clr x ( 2, )

(v)?

9x

xse`9 6 clr x

?

,2

(72) Qse`m c(x) 4 x6/x, eha`tecy wnegn es ifrmar a lr a.

F`swar ; (72) a



DFXNW


Gl`gfve ty, gl`vaxe ty, ple`t lc e`c iagt el`

F cu`g tel` c(x ) e s gl`gfva e` (f, o) ec tf`ma`t hrfw` ft avary ple`t (x2,(c(x

2)), clr x

2 (f, o) iea oailw tna

gurva. c(x) es gl`vax e` (f,o) ec tf`ma`t hrfw` ft afgn ple`t (x2, c(x

2)), x

2 (f, o) iea folva tna gurva.

F ple`t (g, c(g )) lc tna mrfpn y 4 c(x) es sfeh tl oa f ple`t lc e`ci agtel` lc tna mrfpn, ec c(x ) es gl`gfva e`

(g ‗ , g) f`h gl`vax e` (g, g + ) (lr vega varsa), clr slda _+.

_asuits ; 6. Ec c (x) 0 2 x (f, o), tna` tna gurva y 4 c(x) es gl`gfva e` (f, o)

?. Ec c (x) 9 2 x (f, o) tna` tna gurva y 4 c(x) es gl`vax e` (f, o)

7. Ec c es gl`te ulus ft x 4 g f`h c (x) nfs lpplseta sem`s l` aetnar sehas lc g, tna` tna

ple`t (g, c(g)) es f ple`t lc e`ciagtel` lc tna gurva

8. Ec c (g) 4 2 f`h c (g) 2, tna` tna ple`t (g, c(g)) es f ple`t lc e`ciagtel`

Rrlv e`m Eàqufi et eas use`m gurvftura ;

Maàrfiiy tnasa eàqufieteas e`vliva gldpfresl` oatwaa` vfiuas lc twl cu`gtel`s ft slda pfrteguifr

ple`ts.

Axfdpia # 75 ; Rrlva tnft clr f`y twl ùdoars x6

& x?

,7

aa? ?6 xx 0 7

xx? ?6

a

Wliutel` ; Fssuda c(x) 4 ax f`h iat x6

& x?

oa twl ple`ts l` tna gurva y 4 a x.

Iat _ oa f`ltnar ple t wnegn hevehas RV e` rftel 6 ; ?.



DFXNW


y gllrhe`fta lc ple`t _ es7

aa? ?6 xx f`h y gllrhe`fta lc ple`t W es 7

xx? ?6

a

. We`ga c(x) 4 ax es

gl`gfva up, tna ple`t _ weii fiwfys oa folva tna ple`t W.

7

aa? ?6 xx 0 7

xx? ?6

a

Fitar`ft a ; Folva eàqufiety g luih fisl oa afseiy prlvah use`m FD f`h MD.

Axfdpia # 82 ; Ec 2 9 x6

9 x?

9 x7

9 tna` prlva tnft se`

7

xxx 7?60

7

xse`xse`xse` 7?6 . Na`ga

prlva tnft ; ec F, O, G fra f`mias lc f tref`mia tna` dfxedud vfiua lc

se`F + seÒ + se`G es?

77.

Wliutel` ;

Rle`t F, O, G clrd f tref`mia.

y gllrhe`fta lc ga`trleh M es7

xse`xse`xse` 7?6 f`h y gllrhe`fta lc ple`t C es

se`

7

xxx 7?6.

Na`ga se`

7

xxx 7?6

7

xse`xse`xse` 7?6 .

Ec F + O + G 4 , tna`

se`

7

GO F

7

GseÒse` Fse` se`

7

7

GseÒse` Fse`

?

77 GseÒse` Fse`

dfxedud vfiua lc (se`F + seÒ + se`G) 4?

77

Axfdpia # 86 ; Ce`h tna ple`ts lc e`ciagtel` lc tna cu`gtel` c(x) 4 se`?x x Z2, ?T

Wliutel` ; c(x) 4 se`?x

c (x) 4 se`?x

c (x) 4 ? gls?x

c (2) 4 2 x 48

,

8

7



DFXNW


oltn tnasa ple`ts fra e`ciagtel` ple`ts fs sem` lc c (x) gnf`ma l` aetnar sehas lc tnasa ple`ts.

Axfdpia # 8? ; Ce`h tna e`ciagtel` ple`t lc c(x) 4 7x8 ‗ 8x7. Fisl hrfw tna mrfpn lc c(x) meve`m hua edplrtf`ga tl

gl`gfvety f`h ple`t lc e`ciagtel`.

Wliutel` ; c(x) 4 7x8 ‗ 8x7

c (x) 4 6?x7 ‗ 6?x?

c (x) 4 6?x? (x ‗ 6)

c (x) 4 6?(7x? ‗ ?x)

c (x) 4 6?x(7x ‗ ?)

c (x) 4 2 x 4 2,

7

?.

Fmfe ` ax fde è`m sem` lc c (x)

tnus x 4 2,7

?fra tna e`ciagtel` ple`ts

Na`ga tna mrfpn lc c(x) es


(76) Eha`tecy wnegn es mraftar a

a6 ?lr

?6

(7?) Ec 2 9 x6

9 x?

9 x7

9 , tna` prlva tnft se`

8

xxx? 7?60

8

xse`xse`xse`? 7?6

(77) Ec c(x) es dl`ltlègfiiy hagrafse`m cu`gtel` f`h c (x) 0 2. Fssude`m c ‗6(x) axests prlva tnft

?

)x(c )x(c ?6

66

0 c ‗6

?

xx ?6.

F`swar ; (76)a

a6 ?

Milofi Dfxedud ; F cu`gtel` c(x) es sfeh tl nfva milofi dfxedud l` f sat A ec tnara axests ft iafst là g A sugn tnft c(x) c(g)

clr fii x A.

]a sfy milofi dfxedud lggurs ft x 4 g f`h milofi dfxedud (lr milofi dfxedud vfiua) es c(g).

Ilgfi Dfxedf ; F cu`gtel` c(x) es sfeh tl nfva f ilgfi dfxedud ft x 4 g ec c(g) es tna mraftast vfiua lc tna cu`gtel` e` f sdfii

àemnolurnllh (g ‗ n, g + n), n 0 2 lc g.

e.a. clr fii x (g ‗ n, g + n), x g, wa nfva c(x) c(g).e.a. c(g ‗ ) c(g) c(g + ), 2 9 n



DFXNW

"dfèsnjudfrpnysegs.e`" ?=

@lta ; Ec x 4 g es f olu`hfry ple t tna` gl`sehar (g ‗ n, g) lr (g, g + n) (n 0 2) fpprlpreftaiy.

Milofi De` edud ;

F cu`gtel` c(x) es sfeh tl nfva f milofi deèdud l` f sat A ec tnara axests ft iafst là g A sugn tnft c(x) c(g) clr fii x A.

Ilgfi Deèdf ;

F cu`gtel` c(x) es sfeh tl nfva f ilgfi deèdud ft x 4 g ec c(g) es tna iafst vfiua lc tna cu`gtel` e` f sdfii

àemnolurnllh (g ‗ n, g + n), n 0 2 lc g.

e.a. clr fii x (g ‗n, g + n), x g, wa nfva c(x) c(g).

e.a. c(g ‗ ) c(g) c(g + ), 2 9 n

Axtradf ;

F dfxedf lr f deèdf es gfiiah f` axtradf.

Axpif`ftel` ;

Gl`sehar mrfpn lc y 4 c(x), x Zf, oT

x 4 f, x 4 g?, x 4 g

8 fra ple`ts lc ilgfi dfxedf, wetn dfxedud vfiuas c(f), c(g

?), c(g

8) raspagtevaiy.

x 4 g6, x 4 g

7, x 4 o fra ple`ts lc ilgfi deèdf, wetn deèdud vfiuas c(g

6), c(g

7), c(o) raspagtevaiy

x 4 g? es f ple`t lc milofi dfxedud

x 4 g7 es f ple`t lc milofi deèdud

Gl`sehar tna mrfpn lc y 4 n(x), x Zf, o)

f g6

g6

g?

g7

g8 o

n(f)

n(g )?

n(g )6

n(g )8

n(g )7

x 4 g6, x 4 g

8 fra ple`ts lc ilgfi dfxedf, wetn dfxedud vfiuas n(g

6), n(g

8) raspagtevaiy.

x 4 f, x 4 g? fra ple`ts lc ilgfi deèdf, wetn deèdud vfiuas n(f), n(g

?) raspagtevaiy.

x 4 g7 es àetnar f ple`t lc dfxedf `lr f deèdf.

Milofi dfxedud es n(g8)

Milofi deèdud es n(f)

Axfdpia # 87; Iat c(x) 4

2x6

?|x|2|x|. Axfdeà tna oanfvelur lc c(x) ft x 4 2.

Wliutel` ; c(x) nfs ilgfi dfxedf ft x 4 2 (saa cemura).



DFXNW

"dfèsnjudfrpnysegs.e`" ?>

Axfdpia # 88; Iat c(x) 4

7x67x?

6x2)?o7o(

)6ooo(x ‗

?

?77

Ce`h fii plsseoia vfiuas lc o sugn tnft c(x) nfs tna sdfiiast vfiua ft x 4 6.

Wliutel`. Wugn prloiads gf` afseiy oa slivah oy mrfpnegfi fpprlfgn (fs e` cemura).

Na`ga tna iedete`m vfiua lc c(x) crld iact lc x 4 6 snluih oa aetnar mraftar lr aqufi tl tna vfiua lc cu`gtel`

ft x 4 6.

6xied

c(x) c(6)

‗ 6 +)?o7o(

)6ooo(?

?7

‗ 6

)?o()6o(

)6o)(6o( ?

2

o (‗ ?, ‗6) Z6, + )


(78) E` afgn lc cliilwe`m mrfpns eha`tecy ec x 4 f es ple t lc ilgfi dfxedf, deèdf lr àetnar

(e) (ee)

(eee)

(71) Axfdeà tna mrfpn lc cliilwe m cu`gtel`s e` afgn gfsa eha`tecy tna ple`ts lc milofi dfxedud/

deèdud f`h ilgfi dfxedud / deèdud.

(e) (ee) (eee)



DFXNW


F`swars ; (78) (e) Dfxedf (ee) @aetnar dfxedf lr deèdf

(eee) Deèdf

(71) (e) Ilgfi dfxedf ft x 4 ?, Ilgfi deèdf ft x 4 7, Milofi dfxedud ft x 4 ?. @l

milofi deèdud

(ee) Ilgfi de edf ft x 4 ‗ 6, @l ple`t lc Milofi deèdud, `l ple`t lc ilgfi lr

Milofi dfxedf

(eee) Ilgfi & Milofi dfxedud ft x 4 6, Ilgfi & Milofi de edud ft x 4 2.

Dfxedf, Deèdf clr h ec cara`t efoia cu`gt el`s ;

Dara haceètel` lc dfxedf, deèdf oagldas tahelus e` slive`m prloiads. ]a usa harevfteva fs f tlli tl

lvarglda tnes hecceguity.

F àgassfry gl`hetel` clr f` axtradf ;

Iat c(x) oa heccara`tefoia ft x 4 g.

Xnalrad ; F àgassfry gl`hetel` clr c(g) tl oa f` axtradud lc c(x) es tnft c (g) 4 2.

e.a. c(g) es axtradud c (g) 4 2

@lta ; c (g) 4 2 es lìy f àgassfry gl`hetel` out `lt succegea t

e.a. c (g) 4 2 c(g) es axtradud.

Gl`sehar c(x) 4 x7

c ( 2 ) 4 2

out c(2) es lt f` axtradud (saa cemura).

Wucceg ea`t gl`het el` clr f` axtradf ;

Iat c(x) oa f heccara`tefoia cu`gtel .

Xnalrad ; F succegea`t gl`hetel` clr c(g) tl oa f` axtradud lc c(x) es tnft c (x) gnf`mas sem` fs x pfssas tnrlumn g.

e.a. c(g)es f` axtradf (saacemura) c (x) gnf`mas sem` fs x pfssas tnrlumn g.

x 4 g es f ple`t lc dfxedf. c (x) gnf`mas sem` crld plseteva tl àmfteva.

x 4 g es f ple`t lc ilgfi deèdf (saa cemura), c (x) gnf`mas sem` crld àmfteva tl plseteva.

Wtft el`fry ple`ts ;

Xna ple ts l` mrfpn lc cu`gtel` c(x) wnara c (x) 4 2 fra gfiiah stftel`fry ple`ts.

_fta lc gnf`ma lc c(x) es zarl ft f stftel`fry ple`t.



DFXNW


Axfdpia # 81; Ce`h stftel`fry ple`ts lc tna cu`gtel` c(x) 4 8x7 ‗ 3x? ‗ ?8x + 5.

Wliutel` ; c (x) 4 6?x? ‗ 6?x ‗ ?8

c (x) 4 2 x 4 ‗ 6, ?

c(‗ 6) 4 ?7, c(?) 4 ‗ 76

(‗ 6, ?7), (?, ‗ 76) fra stftel`fry ple`ts

Axfdpia # 83; Ec c(x) 4 x7 + fx? + ox + g nfs axtrada vfiuas ft x 4 ‗ 6 f`h x 4 7. Ce`h f, o, g.

Wliutel`. Axtrada vfiuas ofsegfiiy daf` dfxedud lr de edud vfiuas, se ga c(x) es heccara tefoia cu`gtel` slc (‗ 6) 4 2 4 c (7)

c (x) 4 7x? + ?fx + o

c (7) 4 ?= + 3f + o 4 2

c (‗ 6) 4 7 ‗ ?f + o 4 2

f 4 ‗ 7, o 4 ‗ 5, g _

Cerst Harevfteva Xast ;

Iat c(x) oa gl teùlus f`h heccara`tefoia cu`gtel`.

Wtap - Ce h c (x)

Wtap - . Wliva c (x) 4 2, iat x 4 g oa f sliutel`. (e.a. Ce`h stftel`fry ple ts)

Wtap - . Losarva gnf`ma lc sem`

(e) Ec c (x) gnf`mas sem` crld amfteva tl plseteva fs x grlssas g crld iact tl remnt tna` x 4 g es f ple t lc

ilgfi deèdf

(ee) Ec c (x) gnf`mas sem` crld plseteva tl àmfteva fs x grlssas g crld iact tl remnt tna` x 4 g es f ple t lc

ilgfi dfxedf.

(eee) Ec c (x) hlas `lt gnf`mas sem` fs x grlssas g tna` x 4 g es àetnar f ple`t lc dfxedf `lr deèdf.

Axfdpia # 8=; Ce`h tna ple`ts lc dfxedf lr deèdf lc c(x) 4 x? (x ‗ ?)?.

Wliutel`. c(x) 4 x? (x ‗ ?)?

c (x) 4 8x (x ‗ 6) (x ‗ ?)

c (x) 4 2 x 4 2, 6, ?

axfdeè`m tna sem` gnf`ma lc c (x)

Na`ga x 4 6 es ple`t lc dfxedf, x 4 2, ? fra ple`ts lc deèdf.

@lta ; E` gfsa lc gl`teùlus cu`gtel`s ple`ts lc dfxedf f`h deèdf fra fitar`fta.

Axfdpia # 8>; Ce`h tna ple`ts lc dfxedf, deèdf lc c(x) 4 x 7 ‗ 6?x. Fisl hrfw tna mrfpn lc tnes cu`gtel`s.

Wliutel`. c(x) 4 x7 ‗ 6?x

c (x) 4 7(x? ‗ 8) 4 7(x ‗ ?) (x + ?)

c (x) 4 2 x 4 ¶ ?

Clr trfge`m tna mrfpn iat us ce`h dfxedud f`h deèdud vfiuas lc c(x).

63?

63?

)x(c x



DFXNW


Axfdpia # 85 ; Wnlw tnft c(x) 4 (x 7 ‗ 3x ? + 6?x ‗ >) hlas `lt nfva f`y ple`t lc ilgfi dfxedf lr deèdf. Na`ga

hrfw mrfpn

Wliutel`. c(x) 4 x7 ‗ 3x? + 6?x ‗ >

c (x) 4 7(x? ‗ 8x + 8)

c (x) 4 7(x ‗ ?) ?

c (x) 4 2 x 4 ?

out giafriy c (x) hlas `lt gnf`ma sem` folut x 4 ?. c (?+) 0 2 f`h c (? ‗) 0 2. Wl c(x) nfs `l ple`t

lc dfxedf lr deèdf. E` cfgt c(x) es f dl`ltlègfiiy e`grafse`m cu`gtel` clr x _.

Axfdpia # 12 ; Iat c(x) 4 x7 + 7(f ‗ =)x? + 7(f? ‗ 5) x ‗ 6. Ec c(x) nfs plset eva ple`t lc dfxedf, tna` ce`h plss eoia

vfiuas lc 'f'.

Wliutel`. c (x) 4 7 Zx? + ?(f ‗ =)x + (f? ‗ 5)TIat , oa rllts lc c (x) 4 2 f`h iat oa tna sdfiiar rllt. Axfdeè`m sem` gnf`ma lc c (x).

Dfxedf lggurs ft sdfiiar rllt wnegn nfs tl oa plseteva. Xnes ofsegfiiy edpieas tnft oltn

rllts lc c (x) 4 2 dust oa plseteva f`h heste`gt.

(e) H 0 2 f 9=

?5

(ee) ‗f?

o0 2 f 9 =

(eee) c (2) 0 2 f (‗ , ‗ 7) (7, )

crld (e), (ee) f`h (eee) f (‗ , ‗ 7)

=

?5,7


(73) Ce`h tna ple`ts lc ilgfi dfxedf lr deèdf lc cliilwe`m cu`gtel`s

(e) c(x) 4 (x ‗ 6)7 (x + ?)?

(ee) c(x) 4 x7 + x? + x + 6.

F`swar ; 73. (e) Dfx edf ft x 4 ‗ ?, De edf ft x 4 ‗ 1

8

(ee) @l ple`t lc ilgfi dfxedf lr de edf.

Dfxedf, Deèdf clr gl`t eùlus cu`gtel`s ;

Iat c(x) oa f gl teùlus cu`gtel .

Gre tegfi ple`ts ;

Xna ple`ts wnara c (x) 4 2 lr c(x) es `lt heccara`tefoia fra gfiiah gretegfi ple`ts.

Wtftel`fry ple`ts Gretegfi ple`ts.



DFXNW

"dfèsnjudfrpnysegs.e`" 7?

Axfdpia # 16 ; Ce`h gretegfi ple`ts lc c(x) 4 dfx (se`x, glsx) , x (2, ?).

Wliutel` ;

Crld tna cemura et es giafr tnft c(x) nfs tnraa gretegfi ple`ts x 48

,

?

,

8

1.

Edplrtf`t @lta ;

Clr c(x) haceàh l` f suosat lc _, ple`ts lc axtradf (ec axests) lggur ft gretegfi ple`ts

Axfdpia # 1? ; Ce`h tna plsseoia ple`ts lc Dfxedf/Deèdf clr c(x) 4 |x?

‗ ?x| (x _)

Wliutel`. c(x) 4

2xx?x

?x2xx?

?xx?x

?

?

?

c (x) 4

2x)6x(?

?x2)x6(?

?x)6x(?

c (x) 4 2 ft x 4 6 f`h c (x) hlas `lt axest ft x 4 2, ?. Xnus tnasa fra gretegfi ple`ts.

Axfdpia # 17 ; Iat c(x) 4

2xxse`7

2xx62xx ?7

. Axfdeà tna oanfvelur lc c(x) ft x 4 2.

Wliutel`. c(x) es gl`te ulus ft x 4 2.

c (x) 4

2xxgls7

2x62x?x7 ?

c (2+) 4 7 f`h c (2 ‗) 4 ‗ 62 tnus c(x) es `l`-heccara`tefoia ft x 4 2 x 4 2 es f gretegfi ple`t.

Fisl harevfteva gnf` mas sem` crld àmfteva tl plseteva, sl x 4 2 es f ple`t lc ilgfi deèdf.

Axfdpia # 18 ; Ce`h tna gretegfi ple`ts lc tna cu`gtel` c(x) 4 8x7 ‗ 3x? ‗ ?8x + 5 ec (e) x Z2, 7T (ee) x Z‗7, 7T

(eee) x Z‗ 6, ?T.

Wliutel`. c (x) 4 6?(x? ‗ x ‗ ?)

4 6?(x ‗ ?) (x + 6)

c (x) 4 2 x 4 ‗ 6 lr ?

(e) ec x Z2, 7T , x 4 ? es gretegfi ple`t.

(ee) ec x Z‗ 7, 7T, tna` wa nfva twl gretegfi ple`ts x 4 ‗ 6, ?.

(eee) Ec x Z‗ 6, ?T, tna` `l gretegfi ple`t fs oltn x 4 6 f`h x 4 ? oaglda olu`hfry ple ts.

@lta ; Gretegfi ple`ts fra fiwfys e`tarelr ple`ts lc f` e`tarvfi.



DFXNW


Milofi axtradf clr gl`teùlus cu`gtel`s ;

(e) Cu`gtel` hace ah l` gilsah e tarvfi

Iat c(x), x Zf, oT oa f gl`teùlus cu`gtel`

Wtap - E ; Ce`h gretegfi ple`ts. Iat et oa g6, g

? ......., g

`

Wtap - EE ; Ce`h c(f), c(g6).........., c(g

`), c(o)

Iat D 4 dfx½ { c(f), c(g6),..........., c(g

`), c(o)}

d 4 de` ½ {c(f), c(g6), ........c(g`), c(o)}Wtap - D es milofi dfxedud.

d es milofi deèdud.

(ee) Cu`gtel` hace ah l` lpaè`tarvfi.

Iat c(x), x (f, o) oa gl`teùlus cu`gtel`.

Wtap - E Ce`h gretegfi ple ts . Iat et oa g6, g

?, .......g

`

Wtap - EE Ce`h c(g6), c(g

?), ........., c(g

`)

Iat D 4 dfx ½ {c(g6), .......c(g

`)}

d 4 de`½ {c(g6),............,c(g

`)}

Wtap - EEE fxIed c(x) 4

6 (sfy), ‗ox

Ied

c(x) 4 ? (sfy).

Iat 4 de`. {6,

?}, I 4 dfx. {

6,

?}

Wtap -ES

(e) Ec d tna` d es milofi deèdud

(ee) Ec d 0 tna` c(x) nfs `l milofi deèdud

(eee) Ec D I tna` D es milofi dfxedud

(ev) Ec D 9 I , tna` c(x) nfs `l milofi dfxedud

Axfdpia # 11 ; Ce`h tna mraftast f`h iafst vfiuas lc c(x) 4 x 7 ‗ 6?x x Z‗ 6, 7T

Wliutel`. Xna plsseoia ple`ts lc dfxedf/deèdf fra gretegfi ple`ts f`h tna olu`hfry ple`ts.

clr x Z‗ 6, 7T f`h c(x) 4 x 7 ‗ 6?x

x 4 ? es tna lìy gretegfi ple`t.

Axfdeè`m tna vfiua lc c(x) ft ple`ts x 4 ‗6, ?, 7. ]a gf` ce`h mraftast f`h iafst vfiuas.

57

63?

666

)x(c x

Deèdud c(x) 4 ‗ 63 & Dfxedud c(x) 4 66.


(7=) Iat c (x ) 4 ?x7 ‗ 5x? + 6?x + 3

(e) Ce`h tna plsseoia ple ts lc Dfxedf/Deèdf lc c(x) clr x _.

(ee) Ce`h tna ùdoar lc gretegfi ple`ts lc c(x) clr x Z2, ?T.(eee) Hesguss fosliuta (milofi) dfxedf/de edf vfiua lc c(x) clr x Z2, ?T

(ev) Rrlva tnft clr x (6, 7), tna cu`gtel` hlas `lt nfs f Milofi dfxedud.

F`swars ;

(7=) (e) x 4 6, ? (ee) là

(eee) c(2) 4 3 es tna milofi deèdud, c(6) 4 66 es milofi dfxedud

Axfdpia # 13 ; Iat c(x) 4

?

7x2:

?

6xilm

2x:

2x6:xx

?/6

?



DFXNW


Hesguss milofi dfxedf, deèdf clr 4 2 f`h 4 6. Clr wnft vfiuas lc hlas c(x) nfs milofi

dfxedf

Wliutel` ; Mrfpn lc y 4 c(x) clr 4 2

7/? ‗6/?

‗6

‗6

6

@l milofi dfxedf, de edf

Mrfpn lc y 4 c(x) clr 4 6

7/? ‗6/?

‗6

‗6

6

Milofi dfxedf es 6, wnegn lggurs ft x 4 2Milofi deèdf hlas `lt axests

2xIed c(x) 4 2, 2x

Ied c(x) 4 6, c(2) 4

Clr milofi dfxedf tl axests

c(2) 6 6.

Axfdpia # 1= ; Ce`h axtradf lc c(x) 4 7x8 + >x7 ‗ 6>x? + 32. Hrfw mrfpn lc m(x) 4)x(c

82f`h gldda`t l` ets ilgfi

f`h milofi axtradf.

Wliutel` ; c (x) 4 2

6?x (x? + ?x ‗ 7) 4 2 6?x (x ‗ 6) (x + 7) 4 2

x 4 ‗7, 2, 6

c (x) 4 6?(x + 7) x(x ‗ 6)

‗7 2 6 ‗

++

‗

ilgfi deèdf lggurs ft x 4 ‗7, 6

ilgfi dfxedf lggurs ft x 4 2

c(‗7) 4 ‗ =1, c(6) 4 17 fra ilgfi deèdf

c(2) 4 32 es ilgfi dfxedf

xIed c(x) 4 , x

Ied c(x) 4

Na`ga milofi dfxedf hlas `lt axests ; Milofi deèdf es ‗ =1

m(x) 4 ?))x(c (

82c (x)

m(x) nfs sfda gretegfi ple ts fs tnft lc c(x).

F rlumn sjatgn lc y 4 c(x) es

17

32

‗ =1

‗7

6

Iat zarls lc c(x) oa ,



DFXNW


m(), m() fra u`haceàh,

xIed m(x) 4 ,

xIed m(x) 4 ‗ ,

xIed m(x) 4 ‗ , x

Ied m(x) 4

x 4 , x 4 fra fsydptltas lc y 4 m(x).

xIed m(x) 4 2, x

Ied m(x) 4 2

y 4 2 es fisl f` fsydptlta.

x 4 ‗7, x 4 6 fra ilgfi deèdf lc

y 4 c(x) x 4 ‗ 7, x 4 6 fra ilgfi dfxedf lc y 4 m(x)sedeifriy, x 4 2 es ilgfi deèdf lc y 4 m(x)

Milofi axtradf lc m(x) hlas `lt axests.

F rlumn sjatgn lc y 4 m(x) es

‗(>/61)

‗7

6

82/176

?/7


(7>) Iat c (x ) 4?

x+

x

?. Ce`h ilgfi dfxedud f`h ilgfi deèdud vfiua lc c(x). Gf` ylu axpife` tnes

hesgrapf`gy lc ilgfiiy deèdud vfiua oae`m mraftar tnf` ilgfiiy dfxedud vfiua.

(75) Ec c( x) 4

2xxgls

2x)x( ?

, ce`h plsseoia vfiuas lc sugn tnft c(x) nfs ilgfi dfxedf ft x 4 2.

F`swars ; (7>) Ilgfi dfxedf ft x 4 ‗ ?, c(‗?) 4 ‗ ?: Ilgfi de edf ft x 4 ?, c(?) 4 ?.

(75) Z ‗6, 6)

Dfxedf, Deèdf oy nemnar lrhar harevftevas ;

Wagl`h har evft eva tast ;

Iat c(x) nfva harevftevas up tl sagl`h lrhar

Wtap - E. Ce`h c (x)

Wtap - EE. Wliva c (x) 4 2. Iat x 4 g oa f sliutel`

Wtap - EEE. Ce`h c (g)Wtap - ES.

(e) Ec c (g) 4 2 tna` curtnar e`vastemftel` es raquerah

(ee) Ec c (g) 0 2 tna` x 4 g es f ple`t lc deèdf.

(eee) Ec c (g) 9 2 tna` x 4 g es f ple`t lc dfxedf.

Clr dfxedf c (x) gnf`mas crld plseteva tl àmfteva (fs snlw` e` cemura). c (x) es hagrafse`m na`ga c (g) 9 2



DFXNW


Axfdpia # 1> ; Ce`h tna ple`ts lc ilgfi dfxedf lr deèdf clr c(x) 4 se`?x ‗ x, x (2, ).

Wliutel`. c(x) 4 se`?x ‗ x

c (x) 4 ?gls?x ‗ 6

c (x) 4 2 gls ?x 4?

6 x 4

3

,

3

1

c (x) 4 ‗ 8 se` ?x

c

3 9 2 Dfxedf ft x 4

3

c

3

10 2 Deèdf ft x 4

3

1


(82) Ce`h tna ple`ts lc ilgfi dfxedf lr de edf lc c(x) 4 se` ?x ‗ x

(86) Iat c(x) 4 se`x (6 + glsx) : x (2, ?). Ce`h tna ùdoar lc gretegfi ple`ts lc c(x). Fisl eha`tecy

wnegn lc tnasa gretegfi ple`ts fra ple ts lc Dfxedf/De edf.

F`swar ; 82 Dfx edf ft x 4 ` +3 : Deèdf ft x 4 ` ‗

3

86. Xnraa

x 47

es ple`t lc dfxedf.

x 4 es `lt f ple`t lc axtradf.

x 47

1es ple`t lc deèdf.

`tn Harevfteva tast ;

Iat c(x)nfva harevftevas uptl `tn lrhar

Ec c (g) 4 c (g) 4 ..........4 c `‗6(g) 4 2 f`h

c `(g) 2 tna` wa nfva cliilwe`m plsseoeieteas

(e) ` es ava`, c (`)(g) 9 2 x 4 g es ple`t lc dfxedf

(ee) ` es ava`, c (`)(g) 0 2 x 4 g es ple`t lc deèdf.

(eee) ` es lhh, c (`)(g) 9 2 c(x) es hagrafse`m folut x 4 g

(ev) ` es lhh, c (`) 0 2 c(x) es e`grafse`m folut x 4 g.

Axfdpia # 15 ; Ce`h ple`ts lc ilgfi dfxedf lr deèdf lc c(x) 4 x1 ‗ 1x8 + 1x7 ‗ 6

Wliutel`. c(x) 4 x1 ‗ 1x8 + 1x7 ‗ 6

c (x) 4 1x? (x ‗ 6) (x ‗ 7)

c (x) 4 2 x 4 2, 6, 7c (x) 4 62x (?x? ‗ 3x + 7)

@lw, c (6) 9 2 Dfxedf ft x 4 6

c (7) 0 2 De edf ft x 4 7

f`h, c (2) 4 2 `h harevfteva tast cfeis

sl, c (x) 4 72 (?x? ‗ 8x + 6)

c (2) 4 72

@aetnar dfxedf `lr deèdf ft x 4 2.

@lta ; Et wfs vary gl`vaèa`t tl gnagj dfxedf/deèdf ft cerst stap oy axfdeè`m tna sem`

gnf`ma lc c (x) `l sem` gnf`ma lc c (x) ft x 4 2

c (x) 4 1x? (x ‗ 6) (x ‗ 7)



DFXNW

"dfèsnjudfrpnysegs.e`" 7=

Fppiegft el` lc Dfxedf, Deèdf ;

Clr f meva` prloiad, f` lokagteva cu`gtel` gf` oa gl`strugtah e` tards lc là pfrfdatar f`h tna`

axtradud vfiua gf` oa avfiuftah oy aqufte m tna heccara`tefi tl zarl. Fs hesgussah e` `tn harevfteva tast

dfxedf/deèdf gf` oa eha`teceah.

Qsacui Clrdu ifa l c Da`surft el` tl _adadoar ;

6. Sliuda lc f guoleh 4

on.

?. Wurcfga fraf lc guoleh 4 ?(o + on + n).

7. Sliuda lc guoa 4 f7

8. Wurcfga fraf lc guoa 4 3f?

1. Sliuda lc f glà 47

6 r ? n.

3. Gurvah surcfga fraf lc glà 4 r ( 4 sif`t naemnt)

=. Gurvah surcfga fraf lc f gyie`har 4 ?rn.

>. Xltfi surcfga fraf lc f gyie`har 4 ?rn + ?r ?.

5. Sliuda lc f spnara 47

8r 7.

62. Wurcfga fraf lc f spnara 4 8r ?.

66. Fraf lc f gerguifr sagtlr 4?

6r ? , wna` es e` rfhef`s.

6?. Sliuda lc f presd 4 (fraf lc tna ofsa) Ý (naemnt).67. Iftarfi surcfga fraf lc f presd 4 (paredatar lc tna ofsa) Ý (naemnt).

68. Xltfi surcfga fraf lc f presd 4 (iftarfi surcfga fraf) + ? (fraf lc tna ofsa)

(@lta tnft iftarfi surcfgas lc f presd fra fii ragtf`mia).

61. Sliuda lc f pyrfdeh 47

6(fraf lc tna ofsa) Ý (naemnt).

63. Gurvah surcfga fraf lc f pyrfdeh 4?

6(paredatar lc tna ofsa) Ý (sif`t naemnt).

(@lta tnft sif`t surcfgas lc f pyrfdeh fra tref`mias).

Axfdpia # 32; Ec tna aquftel` x7 + px + q 4 2 nfs tnraa rafi rllts, tna` snlw tnft 8p 7 + ?=q? 9 2.

Wliutel`; c(x) 4 x7 + px + q, c (x) 4 7x? + p

c(x) dust nfva là dfxedud 0 2 f`h là deèdud 9 2. c (x) 4 2

x 4 ¶ 2p,7

p

c es dfxedud ft x 4 ‗7

pf`h deèdud ft x 4

7

p

c 27

pc 7p ‗



DFXNW

"df`esnjudfrpnysegs.e`" 7>

27

p

7

p?q

7

p

7

p?q

q? +?=

p8 7

9 2, 8p7 + ?=q? 9 2.

Axfdpia # 36 ; Ce`h twl plseteva `udoars x f`h y sugn tnft x + y 4 32 f`h xy7 es dfxedud.

Wliutel`. x + y 4 32

x 4 3 2 ‗ y xy7 4 (32 ‗ y)y7

Iat c(y) 4 (32 ‗ y) y7 : y (2, 32)

clr dfxedeze`m c(y) iat us ce`h gretegfi ple`ts

c ( y) 4 7y? (32 ‗ y) ‗ y7 4 2

c (y) 4 y? (6>2 ‗ 8y) 4 2

y 4 81

c (81+) 9 2 f`h c (81 ‗) 0 2. Na`ga ilgfi dfxedf ft y 4 81.

Wl x 4 61 f`h y 4 81.

Axfdpia # 3? ; _agtf`mias fra e`sgreoah e`seha f sadegergia lc rfheus r. Ce`h tna ragtf`mia wetn dfxedudfraf.

Wliutel`. Iat sehas lc ragtf`mia oa x f`h y (fs snlw` e` cemura).

F 4 xy.

Nara x f`h y fra `lt e`hapa`ha`t vfrefoias f`h fra raiftah oy Rytnlmlrus tnalrad wetn r.

8

x?

+ y? 4 r ? y 48

xr

??

F(x) 4 x8

xr

??

F(x) 4 8

xr x

8

??

Iat c(x) 4 r ?x? ‗8

x 8

: x (2, r)

F(x) es dfx edud wna` c(x ) es dfxedud

Na`ga c (x) 4 x(?r ? ‗ x?) 4 2

x 4 r ?

fisl c (r ? ) 9 2 f`h c (r ? ) 0 2

gl`cerde`m ft c(x) es dfxedud wna` x 4 r ? & y 4

?

r .

F i e t ar Iat us gnllsa gllrhe`fta systad wetn lreme` fs ga`tra lc gergia (fs snlw` e` cemura).

F 4 xy

F 4 ? (rgls) (rse`)

F 4 r ? se`?

?,2

Giafriy F es dfxedud wna` 48



DFXNW


x 4 r ? f`h y 4?

r .

Axfdpia # 37 ; F snaat lc fraf 82 d? es usah tl dfja f` lpa` tf`j wetn squfra ofsa. Ce`h tna heda`sel`s lc

tna ofsa sugn tnft vliuda lc tnes tf`j es dfxedud.

Wliutel`. Iat ia`mtn lc ofsa oa x datar f`h naemnt oa y datar (fs snlw` e` cemura).

S 4 x?y

fmfe` x f`h y fra raiftah tl surcfga fraf lc tnes tf`j wnegn es aqufi tl 82 d ?.

x? + 8xy 4 82

y 4x8

x82 ?x (2, 82 )

S(x) 4 x?

x8x82 ?

S(x) 48

)xx82( 7

dfxedeze`m vliuda,

S(x) 48

)x782( ?4 2 x 4

7

82d

f`h S(x) 4 ‗

?

x7 S

7

829 2.

Gl`cerde`m tnft vliuda es dfxedud ft x 47

82d.

Axfdpia # 38 ; Ec f remnt gerguifr gyie`har es e`sgreoah e` f meva` gl`a. Ce`h tna heda`sel`s lc tna gyie`har

sugn tnft ets vliuda es dfxedud.

Wliutel`. Iat x oa tna rfheus lc gyie`har f`h y oa ets naemnt

v 4 x?y

x, y gf` oa raiftah oy use`m sedeifr tref`mias (fs snlw` e` cemura).

xr y 4

r n

y 4r

n(r ‗ x)

v(x) 4 x?

r

n(r ‗ x) x (2, r)

v(x) 4r

n(rx ? ‗ x7)

v(x) 4r n x (?r ‗ 7x)



DFXNW


v

7

r ?4 2 f`h v

7

r ?9 2

Xnus vliuda es dfxedud ft x 4

7

r ?f`h y 4

7

n.

@lta ; Cliilwe m clrduifa lc vliuda, surcfga fraf lc edplrtf`t sliehs fra vary usacui e` prloiads lc dfxedf

& deèdf.

Axfdpia # 31 ; Fdl`m fii ramuifr squfra pyrfdehs lc vliuda 73 ? gd7. Ce`h heda`sel`s lc tna pyrfdeh

nfve m iafst iftarfi surcfga fraf.

Wliutel`. Iat tna ia`mtn lc f seha lc ofsa oa x gd f`h y oa tna parpa`heguifr naemnt lc tna pyrfdeh

(saa cemura).

S 47

6Ý fraf lc ofsa x naemnt

S 47

6x?y 4 73 ?

y 4?x

?62>

f`h W 4?

6Ý paredatar lc ofsa x sif`t naemnt

4?

6(8x).

out 4 ??

y8

x

W 4 ?x ??

y8

x 4 ??8 yx8x

W 4

?

?

?8

x

?62>x8x

W(x) 4?

?8

x

)62>.(>x

Iat c(x) 4 x8 +?

?

x

)62>.(>clr deèdeze`m c(x)

c (x) 4 8x7 ‗7

?

x

)62>(634 2

c (x) 4 87

33

x

)3x( 4 2

x 4 3, wnegn f ple`t lc deèdf

Na`ga x 4 3 gd f`h y 4 7 ? .

Axfdpia # 33 ; Iat F(6, ?) f`h O(‗ ?, ‗ 8) oa twl cexah ple`ts. F vfrefoia ple t R es gnlsa` l` tna strfemnt ieà

y 4 x sugn tnft paredatar lc RFO es deèdud. Ce`h gllrhe`ftas lc R.

Wliutel`. We`ga hestf`ga FO es cexah sl clr deèdeze`m tna paredatar lc RFO, wa ofsegfiiy nfva tl

deèdeza (RF + RO)

Iat F oa tna derrlr edfma lc F e` tna ieà y 4 x (saa cemura).

C(R) 4 RF + RO

C(R) 4 RF + ROOut clr RFO



DFXNW

RF + RO FO f`h aqufiety nlih wna` R, F f`h O oagldas gliieàfr. Xnus clr deèdud pftn

ia`mtn ple`t R es tnft spagefi ple`t clr wnegn RF f`h RO oaglda e`geha`t f`h raciagtah rfys

wetn raspagt tl tna derrlr y 4 x.

Aquftel` lc ieà kleè`m F f`h O es y 4 ?x e`tarsagtel` lc tnes ieà wetn y 4 x es tna ple`t R.

Na`ga R (2, 2).

@lta ; Folva gl`gapt es vary usacui oagfusa sugn prloiads oaglda vary ia`mtny oy dfje`m paredatar fs f

cu`gtel` lc plsetel` lc R f`h tna` deèdeze`m et.


(8?) Ce`h tna twl plseteva ùdoars x f`h y wnlsa sud es 71 f`h tna prlhugt x? y1 dfxedud.

(87) F squfra peaga lc te` lc seha 6> gd es tl oa dfha e tl f olx wetnlut tlp oy gutte`m f squfra

crld afgn glràr f`h clihe`m up tna silps tl clrd f olx. ]nft snluih oa tna seha lc tna squfra

tl oa gut lcc sugn tnft vliuda lc tna olx es dfxedud plsseoia.

(88) Rrlva tnft f remnt gerguifr gyie`har lc meva` surcfga fraf f`h dfxedud vliuda es sugn tnft tna

naemnt es aqufi tl tna hefdatar lc tna ofsa.

(81) F `lrdfi es hrfw` tl tna aiiepsa?1

x?

+63

y ?

4 6. Ce`h tna dfxedud hestf`ga lc tnes `lrdfi crld

tna ga`tra.

(83) F ieà es hrfw` pfsse`m tnrlumn ple`t R(6, ?) tl gut plseteva gllrhe`fta fxas ft F f`h O. Ce`h

deèdud fraf lc RFO.

(8=) Xwl tlw`s F f`h O fra setuftah l` tna sfda seha lc f strfemnt rlfh ft hestf`gas f f`h o

raspagtevaiy parpa`heguifrs hrfw` crld F f`h O daat tna rlfh ft ple`t G f`h H raspagtevaiy.Xna hestf`ga oatwaa` G f`h H es g. F nlspetfi es tl oa oueit ft f ple`t R l` tna rlfh sugn tnft

tna hestf`ga FRO es deèdud. Ce`h plsetel` lc R.

F`swars ; (8?) x 4 ?1, y 4 62. (87) 7 gd (81) 6 uèt

(83) 8 uèts (8=) R es ft hestf`ga lc of

fg

crld G.

application of derivatives theory_e

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