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7/21/2019 Application of Derivatives Theory_e
http://slidepdf.com/reader/full/application-of-derivatives-theorye 1/41
DFXNW
"df`esnjudfrpnysegs.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`ltl`egety,
Dfxedf De`edf.
Harevfteva fs rfta lc gnf`ma
E` vfrelus ceaihs lc fppieah dftnadftegs l`a nfs tna quast tl j`lw tna rfta ft wnegn l`a 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
hace`ah fsn
)f(c )nf(c .
Hac e` et el` ; Xna e`stf`tf`alus rfta lc gnf`ma lc c(x) wetn raspagt tl x es hace`ah fs c (x) 42n
ied n
)f(c )nf(c ,
prlvehah tna iedet axests.
@lta ; Xl usa tna wlrh 'e`stf tf`alus', x d fy `lt oa raprasa`te`m teda. ]a usufiiy usa tna wlrh 'rfta lc
gnf`ma' tl daf` 'e`stf`tf`alus 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)
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DFXNW
"df`esnjudfrpnysegs.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 gl`a l` tna mrlu`h
e` sugn f wfy tnft tna naemnt lc tna gl`a es fiwfys l`a - sextn lc rfheus lc ofsa. Nlw cfst es tna
naemnt lc tna sf`h gl`a 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 iaf`e`m fmfe`st f wfii. Xna olttld lc ifhhar es oae`m puiiah fil`m tna
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DFXNW
"df`esnjudfrpnysegs.e`" 7
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 gl`egfi cu``ai 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 ie`a tnrlumn R
parpa`heguifr tl tf`ma`t es gfiiah `lrdfi ft R.
Maldatregfi Daf`e`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 ie`a 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 ie`a y 4 y
6 f`h `lrdfi es tna ie`a 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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DFXNW
"df`esnjudfrpnysegs.e`" 8
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 ie`as wnegn fra tf`ma`t tl gurva y 46x
6
f`h wnegn fra
pfrfiiai tl tna ie`a 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 ie`as (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 `aemnolrnllh 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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DFXNW
"df`esnjudfrpnysegs.e`" 1
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 l`iy l`a rafi vfiua lc n, na`ga tnara es l`iy l`a ple`t lc gl`tfgt edpiye`m tnft l`iy
l`a rafi `lrdfi es plsseoia crld ple`t (6, ?).
Waic prfgtega prloiads ;
(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 ie`as 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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DFXNW
"df`esnjudfrpnysegs.e`" 3
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 ie`a kle`e`m ple`ts (2, 7) f`h (1, ‗ ?) oagldas tf`ma`t tl gurva
y 46x
g
.
Wliutel` ; Aquftel` lc ie`a kle e`m F & O es x + y 4 7
Wlive`m tnes ie`a 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 ie`a tlugnas f gurva tna` l` slive`m tna aquftel` lc ie`a 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 ie`a f`h gurva y 4 x 7 e.a., ple`t R f`h V. (saa
cemura)
Out, se`ga ie`a 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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DFXNW
"df`esnjudfrpnysegs.e`" =
Waic prfgtega prloiads ;
(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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DFXNW
"df`esnjudfrpnysegs.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
Waic prfgtega prloiads ;
(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`tfe`ah
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 hace`ah 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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DFXNW
"df`esnjudfrpnysegs.e`" 5
@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` l`a 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 l`a sliutel`.
Wliutel` ; Xwl gurvas dust e`tarsagt ft l`iy l`a ple`t.
(e) (ee)
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"df`esnjudfrpnysegs.e`" 62
E. Ec p 2 tna` tnara axests l`iy l`a sliutel` (saa mrfpn - (e))
EE. Ec p 0 2
tna` tna twl gurvas dust l`iy 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 hace`ah)
Axfdpia # 6> Ce`h tna snlrtast hestf`ga oatwaa` tna ie`a y 4 x ‗ ? f`h tna pfrfolif y 4 x? + 7x + ?.
Wliutel` ; Iat R(x6, y
6) oa f ple`t gilsast tl tna ie`a y 4 x ‗ ?
tna`)y,x( 66
hx
hy4 silpa lc ie`a
?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
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"df`esnjudfrpnysegs.e`" 66
p 4?
7.
Waic prfgtega prloiad ;
(61) Ce`h tna de`edud & 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 hace`a 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 hace`ah l` Zf, oT es
(e) gl`te`ulus l` Zf, oT
(ee) harevfoia l` (f, o) f`h
(eee) c(f) 4 c(o),
tna` tnara axests ft iafst l`a rafi `udoar 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`te`ulus 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 l`a ple`t wnara tna tf`ma`t es pfrfiiai tl x-fxes.
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"df`esnjudfrpnysegs.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 ftiafstl`a zarl lcc (x)
Axfdpia # 65 ; Sarecy _liia―s tnalrad clr c(x) 4 (x ‗ f)` (x ‗ o)d , wnara d, ` fra plseteva rafi `udoars, clr
x Zf, oT.
Wliutel` ; Oae`m f pliy ldefi cu`gtel` c(x) es gl`te`ulus 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 ‗ (`o + df)T 4 2
x 4`d
df`o
, 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 l`a 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 l`a 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 l`a zarl lc c(x).
(6=) Wnlw tnft clr f`y rafi `udoars , tna pliy`ldefi R(x) 4 x= + x7 + , nfs axfgtiy l`a rafi rllt.
Ifmrf`ma―s Daf` Sfiua Xnalrad IDSX ;
Ec f cu`gtel` c hace`ah l` Zf, oT es
(e) gl`te`ulus l` Zf, oT f`h
(ee) harevfoia l` (f, o)
tna` tnara axests ft iafst l`a rafi `udoars 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 hatarde`ah 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 te`ulus f`h harevfoia cu`gtel`s et saic
(e) gl`te`ulus 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 `udoar 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
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"df`esnjudfrpnysegs.e`" 67
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 l`a 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` `udoar 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
ftf`otf` 66
clr f 9 g 9 o f`h c (x) 4 ?x6
,
@lw c (x) es f dl`ltl`egfiiy hagrafse m cu`gtel`
Na ga ec f 9 g 9 o c (o) 9 c (g) 9 c (f)
?o6
6
9
fo
ftf`otf` 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
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"df`esnjudfrpnysegs.e`" 68
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`ltl`egety 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`ltl`egfiiy
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`ltl`egfiiy 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`ltl`egfiiy e`grafse m (`l` hagrafse`m). Out gl`varsa `aah
`lt oa trua.
(ee) c es stregtiy hagrafse m c es d l`ltl`egfiiy hagrafse`m (`l` e`grafse`m). Fmfe`, gl`varsa `aah
`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`ltl`egfiiy hagrafse`m
(v) c es sfeh tl oa f dl`ltl eg cu`gtel` ec aetnar et es dl`ltl`egfiiy e grafse`m lr dl`ltl`egfiiy
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`ltl`eg
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 }
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Axfdpia # ?8 ; Iat c(x) 4 x7. Ce`h tna e`tarvfis lc dl`ltl`egety.
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`ltl`egfiiy 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`ltl`egety 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 ‗ ?)?
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"df`esnjudfrpnysegs.e`" 63
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`te`ulus 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`ltl`eg
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`ltl`eg 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`ltl`egfiiy 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 _
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"df`esnjudfrpnysegs.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 u`el` lc (e) f`h (ee), wa mat f (‗ , 7T.
Waic prfgtega prloiads ;
(?6) Ce`h tna e`tarvfis lc dl`ltl`egety 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`ltl`egfiiy e`grafse`m clr x _.
(?7) Ec c(x) 4 ?ax ‗ fa ‗x + (?f + 6) x ‗ 7 dl`ltl`egfiiy 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`ltl`egfiiy 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`ltl`egfiiy 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
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DFXNW
"df`esnjudfrpnysegs.e`" 6>
Dl`ltl`egety 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`tfe`e`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`tfe`e`m f (fs snlw` e` cemura).
@lta ; Ec x 4 f es f olu`hfry ple`t tna` usa tna fpprlprefta l`a sehah e`aqufiety tl tast dl`ltl`egety lc c(x).
a.m. ; ]negn lc tna cliilwe`m cu`gtel`s (fs snlw` e` cemura) es e`grafse`m, hagrafse`m lr `aetnar e`grafse`m `lr
hagrafse`m ft
x 4 f.
(e) (ee)
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DFXNW
"df`esnjudfrpnysegs.e`" 65
(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` axfde`a tna sem` lc c (x) l` tna iact `aemnolurnllh f`h tna remnt `aemnolurnllh
lc f.
(e) Ec c (x) es plseteva l` oltn tna `aemnolurnllhs, tna` c es e`grafse`m ft x 4 f.
(ee) Ec c (x) es `amfteva l` oltn tna `aemnolurnllhs, tna` c es hagrafse`m ft x 4 f.
(eee) Ec c (x) nfva lpplseta sem`s l` tnasa `aemnolurnllhs, tna` c es `l`-dl`ltl`eg ft x 4 f.
Axfdpia # 7?; Iat c(x) 4 x7 ‗ 7x + ?. Axfde`a tna dl`ltl`egety 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 `aemnolurnllh f`h c (x) es `amfteva e` remnt `aemnolurnllh.
`aetnar 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 l`iy clr cu`gtel`s tnlsa fra gl`te`ulus ft x 4 f.
Waic prfgtega prloiads ;
(?3) Clr afgn lc tna cliilwe`m mrfpn gldda`t l` dl`ltl`egety lc c(x) ft x 4 f.
(e) (ee) (eee) (ev)
(?=) Iat c (x ) 4 x7 ‗ 7x? + 7x + 8, gldda`t l` tna dl`ltl`eg 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`ltl`eg
oanfvelur lc c(x) ft x 4 2, 6, ?. Es c(x) D. . clr x Z2, ?T <
F`swars ; (?3) (e) `aetnar 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, ? : `aetnar D.. `lr D.H. ft x 4 6. @l, c(x) es `lt D.. clr x Z2, ?T.
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DFXNW
"df`esnjudfrpnysegs.e`" ?2
Qsa lc dl`ltl`ege ty clr p rlve`m e`aquf ie teas
Gldpfresl` lc twl cu`gtel`s c(x) f`h m(x) gf` oa hl`a oy f`fiyse`m tna dl`ltl`eg 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)
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DFXNW
"df`esnjudfrpnysegs.e`" ?6
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 e`aqufieteas, 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 `amfteva, na`ga iat us gnagj clr
dl`ltl`eg `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
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DFXNW
"df`esnjudfrpnysegs.e`" ??
Axfdpia # 73 ; Axfde`a 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`ltl`egfiiy 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.
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DFXNW
"df`esnjudfrpnysegs.e`" ?7
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 axfde`a dl`ltl`eg `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
Waic prfgtega prloiads ;
(?5) Rrlva tna cliilwe`m e`aqufieteas
(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
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DFXNW
"df`esnjudfrpnysegs.e`" ?8
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 E`aqufi et eas use`m gurvftura ;
Ma`arfiiy tnasa e`aqufieteas e`vliva gldpfresl` oatwaa` vfiuas lc twl cu`gtel`s ft slda pfrteguifr
ple`ts.
Axfdpia # 75 ; Rrlva tnft clr f`y twl `udoars 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 ; ?.
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DFXNW
"df`esnjudfrpnysegs.e`" ?1
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 e`aqufiety 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 + se`O + 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
Gse`Ose` Fse` se`
7
7
Gse`Ose` Fse`
?
77 Gse`Ose` Fse`
dfxedud vfiua lc (se`F + se`O + 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
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DFXNW
"df`esnjudfrpnysegs.e`" ?3
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 `e`m sem` lc c (x)
tnus x 4 2,7
?fra tna e`ciagtel` ple`ts
Na`ga tna mrfpn lc c(x) es
Waic prfgtega prloiads ;
(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`ltl`egfiiy 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 l`a 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
`aemnolurnllh (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
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DFXNW
"df`esnjudfrpnysegs.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 de`edud l` f sat A ec tnara axests ft iafst l`a g A sugn tnft c(x) c(g) clr fii x A.
Ilgfi De`edf ;
F cu`gtel` c(x) es sfeh tl nfva f ilgfi de`edud ft x 4 g ec c(g) es tna iafst vfiua lc tna cu`gtel` e` f sdfii
`aemnolurnllh (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 de`edf 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 de`edf, wetn de`edud 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 de`edud
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 de`edf, wetn de`edud vfiuas n(f), n(g
?) raspagtevaiy.
x 4 g7 es `aetnar f ple`t lc dfxedf `lr f de`edf.
Milofi dfxedud es n(g8)
Milofi de`edud es n(f)
Axfdpia # 87; Iat c(x) 4
2x6
?|x|2|x|. Axfde`a tna oanfvelur lc c(x) ft x 4 2.
Wliutel` ; c(x) nfs ilgfi dfxedf ft x 4 2 (saa cemura).
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"df`esnjudfrpnysegs.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, + )
Waic prfgtega prloiads ;
(78) E` afgn lc cliilwe`m mrfpns eha`tecy ec x 4 f es ple t lc ilgfi dfxedf, de`edf lr `aetnar
(e) (ee)
(eee)
(71) Axfde`a tna mrfpn lc cliilwe m cu`gtel`s e` afgn gfsa eha`tecy tna ple`ts lc milofi dfxedud/
de`edud f`h ilgfi dfxedud / de`edud.
(e) (ee) (eee)
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"df`esnjudfrpnysegs.e`" ?5
F`swars ; (78) (e) Dfxedf (ee) @aetnar dfxedf lr de`edf
(eee) De`edf
(71) (e) Ilgfi dfxedf ft x 4 ?, Ilgfi de`edf ft x 4 7, Milofi dfxedud ft x 4 ?. @l
milofi de`edud
(ee) Ilgfi de edf ft x 4 ‗ 6, @l ple`t lc Milofi de`edud, `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, De`edf clr h ec cara`t efoia cu`gt el`s ;
Dara hace`etel` lc dfxedf, de`edf oagldas tahelus e` slive`m prloiads. ]a usa harevfteva fs f tlli tl
lvarglda tnes hecceguity.
F `agassfry gl`hetel` clr f` axtradf ;
Iat c(x) oa heccara`tefoia ft x 4 g.
Xnalrad ; F `agassfry 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 l`iy f `agassfry 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 `amfteva.
x 4 g es f ple`t lc ilgfi de`edf (saa cemura), c (x) gnf`mas sem` crld `amfteva 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.
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"df`esnjudfrpnysegs.e`" 72
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 te`ulus 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 de`edf
(ee) Ec c (x) gnf`mas sem` crld plseteva tl `amfteva 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 `aetnar f ple`t lc dfxedf `lr de`edf.
Axfdpia # 8=; Ce`h tna ple`ts lc dfxedf lr de`edf 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, ?
axfde`e`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 de`edf.
@lta ; E` gfsa lc gl`te`ulus cu`gtel`s ple`ts lc dfxedf f`h de`edf fra fitar`fta.
Axfdpia # 8>; Ce`h tna ple`ts lc dfxedf, de`edf 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 de`edud vfiuas lc c(x).
63?
63?
)x(c x
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Axfdpia # 85 ; Wnlw tnft c(x) 4 (x 7 ‗ 3x ? + 6?x ‗ >) hlas `lt nfva f`y ple`t lc ilgfi dfxedf lr de`edf. 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 de`edf. E` cfgt c(x) es f dl`ltl`egfiiy 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. Axfde`e`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
Waic prfgtega prloiads ;
(73) Ce`h tna ple`ts lc ilgfi dfxedf lr de`edf 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, De`edf clr gl`t e`ulus cu`gtel`s ;
Iat c(x) oa f gl te`ulus 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.
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"df`esnjudfrpnysegs.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) hace`ah 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/De`edf 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
. Axfde`a 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 `amfteva tl plseteva, sl x 4 2 es f ple`t lc ilgfi de`edf.
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.
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Milofi axtradf clr gl`te`ulus cu`gtel`s ;
(e) Cu`gtel` hace ah l` gilsah e tarvfi
Iat c(x), x Zf, oT oa f gl`te`ulus 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 de`edud.
(ee) Cu`gtel` hace ah l` lpa`e`tarvfi.
Iat c(x), x (f, o) oa gl`te`ulus 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 de`edud
(ee) Ec d 0 tna` c(x) nfs `l milofi de`edud
(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/de`edf 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 l`iy gretegfi ple`t.
Axfde`e`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
De`edud c(x) 4 ‗ 63 & Dfxedud c(x) 4 66.
Waic Rrfgtega Rrloiads ;
(7=) Iat c (x ) 4 ?x7 ‗ 5x? + 6?x + 3
(e) Ce`h tna plsseoia ple ts lc Dfxedf/De`edf lc c(x) clr x _.
(ee) Ce`h tna `udoar 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) l`a
(eee) c(2) 4 3 es tna milofi de`edud, c(6) 4 66 es milofi dfxedud
Axfdpia # 13 ; Iat c(x) 4
?
7x2:
?
6xilm
2x:
2x6:xx
?/6
?
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"df`esnjudfrpnysegs.e`" 78
Hesguss milofi dfxedf, de`edf 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 de`edf 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 de`edf lggurs ft x 4 ‗7, 6
ilgfi dfxedf lggurs ft x 4 2
c(‗7) 4 ‗ =1, c(6) 4 17 fra ilgfi de`edf
c(2) 4 32 es ilgfi dfxedf
xIed c(x) 4 , x
Ied c(x) 4
Na`ga milofi dfxedf hlas `lt axests ; Milofi de`edf 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 ,
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m(), m() fra u`hace`ah,
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 de`edf 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 de`edf 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
Waic Rrfgtega Rrloiads ;
(7>) Iat c (x ) 4?
x+
x
?. Ce`h ilgfi dfxedud f`h ilgfi de`edud vfiua lc c(x). Gf` ylu axpife` tnes
hesgrapf`gy lc ilgfiiy de`edud 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, De`edf 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 de`edf.
(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 `amfteva (fs snlw` e` cemura). c (x) es hagrafse`m na`ga c (g) 9 2
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"df`esnjudfrpnysegs.e`" 73
Axfdpia # 1> ; Ce`h tna ple`ts lc ilgfi dfxedf lr de`edf 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 De`edf ft x 4
3
1
Waic prfgtega prloiads ;
(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 `udoar 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 : De`edf 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 de`edf.
`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 de`edf.
(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 de`edf 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 de`edf ft x 4 2.
@lta ; Et wfs vary gl`va`ea`t tl gnagj dfxedf/de`edf ft cerst stap oy axfde`e`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)
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DFXNW
"df`esnjudfrpnysegs.e`" 7=
Fppiegft el` lc Dfxedf, De`edf ;
Clr f meva` prloiad, f` lokagteva cu`gtel` gf` oa gl`strugtah e` tards lc l`a pfrfdatar f`h tna`
axtradud vfiua gf` oa avfiuftah oy aqufte m tna heccara`tefi tl zarl. Fs hesgussah e` `tn harevfteva tast
dfxedf/de`edf 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 gl`a 47
6 r ? n.
3. Gurvah surcfga fraf lc gl`a 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 l`a dfxedud 0 2 f`h l`a de`edud 9 2. c (x) 4 2
x 4 ¶ 2p,7
p
c es dfxedud ft x 4 ‗7
pf`h de`edud ft x 4
7
p
c 27
pc 7p ‗
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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
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DFXNW
"df`esnjudfrpnysegs.e`" 75
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)
7/21/2019 Application of Derivatives Theory_e
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DFXNW
"df`esnjudfrpnysegs.e`" 82
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
& de`edf.
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 de`edeze`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 de`edf
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 ie`a
y 4 x sugn tnft paredatar lc RFO es de`edud. Ce`h gllrhe`ftas lc R.
Wliutel`. We`ga hestf`ga FO es cexah sl clr de`edeze`m tna paredatar lc RFO, wa ofsegfiiy nfva tl
de`edeza (RF + RO)
Iat F oa tna derrlr edfma lc F e` tna ie`a y 4 x (saa cemura).
C(R) 4 RF + RO
C(R) 4 RF + ROOut clr RFO
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DFXNW
RF + RO FO f`h aqufiety nlih wna` R, F f`h O oagldas gliie`afr. Xnus clr de`edud 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 ie`a kle`e`m F f`h O es y 4 ?x e`tarsagtel` lc tnes ie`a 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` de`edeze`m et.
Waic Rrfgtega Rrloiads ;
(8?) Ce`h tna twl plseteva `udoars 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 glr`ar 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 ie`a es hrfw` pfsse`m tnrlumn ple`t R(6, ?) tl gut plseteva gllrhe`fta fxas ft F f`h O. Ce`h
de`edud 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 de`edud. Ce`h plsetel` lc R.
F`swars ; (8?) x 4 ?1, y 4 62. (87) 7 gd (81) 6 u`et
(83) 8 u`ets (8=) R es ft hestf`ga lc of
fg
crld G.