Ejercicios 5.8 Calculus Tom Apostol Vol I
August 1, 2022 | Author: Anonymous | Category: N/A
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6.
∟
5x + 6 ix
Yhafbih kf sustjtunj¹hhb? b? u 1 5x + 6 , ku`mh, iu ix , , y, ix 1 ix 1 ku`mh, iu 1 5ix
iu 5
^ustjtuy`bih?
∟ 5x + 6 ix ∟ u iu 1 6 ∟ u iu 6 iu 1 6 ix ↕ 1 5
∟ 1
5.
6 5
>/5
u > /5
n 1 + n 1
5
5
6/5
u
6 iu iu 1 1 5
6 >/5 6 >/5 u + n 1 n 1 (5 x + 6) + n (5x > >
x 6 + >x >x ix
Yhafbih kf sustjtunj¹hhb? b? u 1 6 + >x > x, ku`mh, iu ku`mh, iu 1 >ix
iu u ∕ 6 ↕ ix ix 1 1 , y, x 1 > >
^ustjtuy`bih?
6
6 5
u +6 6 5 + 6
+ n
∟
x 6 + >x >x ix
1
∟
6
: 6 1 : 6 1 :
∟ ∕ ↕ u
>
∟ u iu
u u iu
>.
1
∟ >
6
u iu
∕ 6)∟ u iu ∟ u iu
∕ ∕ ∕ ∕ ∕ ∕ ∕ :
u>/5 iu > 5
u6/5 iu
6 5
u >/5 + n >/5
u +6 6 u0/5 n 1 n + 1 6 : 0/5 5 + 6
u +6 > 5 + 6
6 6 u0/5 : : 0 /5 5 0/5 1 (6 + >x >x) 90
1
iu 6 u 1 (u > :
6
∕
u>/5 5 0/5 n 1 u + n 1 90 > /5 5 >/5 (6 + >x >x) + n 5=
5 >/5 u + n 5=
∟
x5 x + 6 ix
u 1 x + x 1 u u ^`f u ^`f 1 x + 6, iu 6, iu 1 ix 1 ix,, y, x 1
∕ 6 ↕ x5 1 (u ∕ 6)5 1 u5 ∕ 5u + 6
^ustjtuy`bih?
∟ ↕ ∕ ∟ ∕ ∟ ∟ ∟ ∕ ∟ ∕ ∕ ∕ x5 x + 6 ix 6 ix 5
u
1
u
(u5
5u u +
1
u0
1
u0/5 iu
1
u +6 0 5 + 6
iu
u>
5
5
u iu iu 1 1
iu iu + +
∟
u0
∟
∕5
∟ > ∟ u + u
iu
u iu
iu + 5 u>/5 iu + u6/5 iu > 5
0 5
∟
5u + 6) u iu
u +6 > 5 + 6
6 5
u +6 + 6 + 6 + n 5
u >/5 u=/5 u0/5 5 9 0/5 5 >/5 n 1 u=/5 u + u + n + n 1 + 5 >/5 = 0 > = /5 0/5 5 9 5 1 (x + 6)=/5 (x + 6)0/5 + (x + 6)>/5 + n = 0 > 1
∕
∕
6/>
9.
∕5/>
u 1 ^`f u ^`f
∟ x5 ∕ix>x
∟ 5 ∕ >x ↕ u5 1 5 ∕ >x , ku`mh, 5uiu 5uiu 1 1 ∕>ix 5
5
ix ix 1 1
5 ∕ u ∕ 5> uiu uiu , , y, >x >x 1 5 ∕ u5 ↕ x 1 >
Ifih qu` s` gfn` ubf sustjtunj¹hb, hb, khs k¹Ķajt`s i` kfs jbt`mrfk`s nfaejfb, `sth `s?
∕ 5> ≩ x ≩ 6> ∕5 ≩ >x ≩ 6 Auktjpkjnfbih kf i`sjmufkifi phr 5
∕6
≠ ∕>x ≠ ∕6 ↕ ∕6 ≩ ∕>x ≩ 5
suafbih 5 f kf i`sjmufkifi?
∕6 + 5 ≩ 5 ∕ >x ≩ 5 + 5 6
≩ 5 ∕ >x ≩ 9, ifih qu`, u qu`, u 5 1 5 ∕ >x, s` hetj`b`?
6
≩ u5 ≩ 9, `xtrfy`bih rf¹ rf¹Ķz nufirfi nufirfif? f?
6
≩u≩5
^ustjtuy`bih? 6/>
∕5/>
1
5 :
x ix 5 >x
∟ ∕ 5
5
5 5∕u5 >
5 iu 1 ( )u iu 1 u >
↕ 6
u5 (u) iu 1
∕
5
5 :
u
6
1
0.
5 5u :
u >
5 :
7 >
9
5
6
∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕ ∕
∕ u 5 (∕ 5 )u iu >u
6
>
>
5 :
1
6
5
5(5)
6 >
>
u5 Q iu
\5
> 5
1
5
1
5 >
5(6)
5 9 : >
0 1 >
∕ ∕
∕ 6>
(x + 6) ix (x5 + 5x + 5)>
u 1 x 5 + 5x ^`f u ^`f 1 x 5x + 5 , ku`mh iu ku`mh iu 1 1 (5x (5x + 5)ix 5)ix iu 1 5(x dfnthrjzfbih, iu dfnthrjzfbih, 5(x + 6)ix 6)ix
↕ iu5 1 ((xx + 6)ix 6)ix >
5 :
5 6 1 5= >
^ustjtuy`bih? (x + 6) ix 6) ix 5 5x + 5)> (x + 5x
iu 5 1 u>
↕
6 u ∕>+6 6 u∕5 1 \ Q + n + n 1 1 \ Q + n + n 1 1 5 >+6 5 5
1
4.
∕
iu 6 1 u> 5
iu 6 1 5u> 5
∕
∕
u∕> iu
6 ∕5 u + n 9
6 n 1 ∕ ∕ 69 (x5 + 5x 5x + 5)∕5 + n 1 + n 5 5x + 5) 5 9(x 9(x + 5x
s`b> x ix
>
5
s`b x ix ix 1 ix 1 (6 1 (s`b x)s`bx ix 1
1 \s`bx
5
∕ s`bxnhs xQ ix 1
∕
∕
u 1 nhsx , ^`f u ^`f 1 nhsx , ku`mh, iu ku`mh, iu 1 sustjtuy`bih `b
∕
5
∕s`bx ix
s`bxnhs5 x ix ix s` s` hetj`b`?
5
nhs x( s`bx s`bx))ix
∕
nhs> x u> b 1 b , phr tfbth, + b 1 + b , > >
s`bxnhs5 x ix
∕
s`bxnhs x ix ix 1 1
1
s`bx ix ix + +
∕
s`bxnhs5 x ix ix + l + l
nhsx + nhsx +
1
∕ nhs5x)s`bx ix
>
s`b x ix ix 1 1
∕nhsx nhsx + +
∕
↕
u5 iu iu 1 1
s`bxnhs5 x ix ix + l 1 + l 1
u5+6 + b + b 5+6
l + b 1 n , ifih qu` l qu` l y b shb b¹ u ua`rhs a`rhs r`fk`s, r`fk`s, l + b 1 n , ku`mh,
s`b> x ix ix 1 1
=.
z (z
6 > nhs x >
∕ nhsx nhsx + n + n
∕ 6)6/> iz
u 1 z ^`f u ^`f 1 z
u 1 z z ∕ 6 ↕ z 1 u ∕ 6, ku`mh , iu , iu 1 iz 1 iz , y, u 1 1 u + + 6
^ustjtuy`bih?
9
>
nhs x ∕nhsx nhsx + l + b + + l + b >
z (z
6/>
∕ 6)
iz
↕
(u + 6)u 6)u
6/>
iu iu 1 1 \u(u6/> ) + u + u6/> Q iu
1 \u6/>+6 + u6/> Q iu 1 \u9/> + u6/> Q iu 1
u=/> u 9/> u6/>+6 u9/>+6 n 1 + n + + n + 1 + 1 9/> =/> 9 /> + 6 6 /> + 6 1
7.
> =/> > 9/> > u + u + n 1 n 1 (z = 9 =
u9/> iu iu + + u6/> iu
∕ 6)=/> + >9 (z ∕ 6)9/> + n
nhsx ix s`b> x
^`f u ^`f u 1 1 s`bx s`bx,, ku`mh, iu ku`mh, iu 1 1 nhsx nhsx ix ^ustjtuy`bih? ∕>+6
nhsx s`b>ix x ix
∕5
u∕> iu iu 1 1 u> + 6 + n + n 1 1 u 5 + n
∕
∕
6 6 6 5 n 1 ∕ (s`bx s`bx))∕5 + n 1 n 1 ∕ n 1 nsn x + n ∕ 65 u∕5 + n 1 + n + 1 ∕ + n 5 5s`b5 x 5
1
σ/9 σ/9
:.
iu u> 1
↕
∟ ∕ s`b5x ix
nhs5x 9
3
u 1 9 ^`f u ^`f
iu s`b55x, ku`mh, iu nhs55x ix ↕ ∕ 1 nhs nhs55x ix ∕ s`b ku`mh, iu 1 1 ∕5nhs 5
Ifih qu` s` gfn` ubf sustjtunj¹hb, hb, khs k¹Ķajt`s i` jbt`mrfnj jbt`mrfnj¹hb ¹hb nfaejfb, `sth `s? 3
≩ x ≩ σ9
ifih qu` u qu` u 1 1 9
∕ s`b s`b55x, gfnj`bih x13, u s`b(3) x13, u 1 9 ∕ s`b (3) 1 9
σ , u 1 9 9 Fghrf, sustjtuy`bih?
x 1 gfnj`bih x gfnj`bih
σ/9 σ/ 9
3
∕ s`b
σ 5( ) 1 9 9
>
∟ ∕ s`b s`b5x ix ↕ nhs5x nhs 5
9
5
gfnj`bih ush i` kf prhpj`ifi
∕ s`b
σ 1 9 5
∕61 >
>
∟ ∕ ∕ ∟ ∕ 9
iu u( ) 1 5
6 5
u iu iu 1 1
9
∕
6 5
f
e
d d ((x) ix ix,, s` tj`b`,
d ( d (x) ix 1
e
f
0
>
9
u6/5 iu
9
6 1 5
1 > \7 63.
u
>
6
6/5
6 u 6/5+6 iu iu 1 1 5 6/5 + 6
∟
∕>
∟
7
>Q 1 >
∕
9 >
1
>
6 >/5 u >
9
1
>
6 >/5 9 >
∕ >>/5
s`b s`bx x ix
(> + nhsx)5
s`bx ix ix (> + nhsx + nhsx))5
∕s`bx ix
iu 1 u5
↕ ∕
n 1 (> + nhsx n 1 1 u ∕6 + n 1 + nhsx))∕6 + n 1
∕
∕5
u
iu iu 1 1
6 + n + n > + nhsx + nhsx
∕
u ∕5+6 + n n 1 1 5+6
∕
u∕6 + n 6
∕ ∕
s`bx ix
66.
nhs> x nhs u 1 nhsx, x, iu iu 1 ^`f u ^`f 1 nhs 1
∟
iu 1 s`bx ix ∕s`bx ix ↕ ∕iu 1 s`bx
s`bx ix
iu
∟ ↕ ∕ ∟ nhs> x
u∕6/5
1
6 5
7
u>
>
u∕>/5+6 n 1 + n 1 > 5 + 6
∕
∟
∟ x + 6,6, iu 6 6 ix ↕ 5iu iu 1 ix iu 1 1 ∟ 1 ∟ x + 6 5 x + 6
∟ ∟
s`b x + 6 ix 6 ix x + 6
∟ > + 6 1 5, y, x 1 x 1 7, u 7, u 1 ∟ 7 + 6 1 > >
>
↕
xb∕6 s`bxb ix,
iu 1 bx bxb∕6 ix u 1 x 1 x b , iu 1
u∕6/5
∕ ∕ 65
5 5 5 n 1 n n n 1 + 1 + 1 + n + n nhsx u6/5 nhsx))6/5 (nhsx
iu 1 5 5s`bu iu 1
5
5
∕5( 5(nhs nhs>> ∕ nhs nhs5) 5) 1 5(nhs 5(nhs 5 5 ∕ nhs nhs >) >)
∕
∟ ∟ x + 6
x 1 >, u gfnj`bih x gfnj`bih >, u 1 1 7
iu 1 u∕>/5 iu 1
n 1 5u∕6/5 + n 1 n 1 + n 1
u 1 Yhafbih u Yhafbih 1
1
∕
s`b x + 6 ix
>
6>.
9
>
u 1 nhsx, x, iu 1 iu 1 Yhafbih u Yhafbih 1 > + + nhs
65.
6 u>/5 1 5 >/5
b13
↕ iu 1 xb∕6 ix b 4
s`bu iu 1 iu 1
nhsuQQ>5 ∕5 \nhsu
+ n
sustjtuy`bih xb∕6 s`bxb ix
s`bu 6 s`bu iu iu iu 1 1 b b nhsxb 6 b nhsx + n 1 n 1 + n b b
↕ 6 nhsu + n 1 1 ∕ nhsu + n 1 ∕ b
69.
∕
x0 ix
∟
∕ x4
6
^`f u 1 6 ^`f u
iu 1 x 0 ix ∕ x4, iu 1 ∕4x0 ix ↕ ∕ 64 iu 1 x
x0 ix 6 6 iu iu 1 u∕6/5 iu 1 1 4 u 4 4 6 x 6 6 6 n 1 \(6 x4 )6/5 Q + n + n 1 1 5u6/5 + n 1 1 > > 4
∟ ∕
60.
∕ ↕ ∕
∟ ∕
∕
∕
∕
∕
6 u∕6/5+6 + n 1 6 4 5 + 6
∕ 6
∕
6 u6/5 +n 6 4 5
∕
x4 + n
t(6 + t)6/9 it
u 1 6 + t iu 1 it it + t , , iu 1 u 1 6 + t + t
↕ u ∕ 6 1 t
sustjtuy`bih
∕ ↕ ∕ ∕ ∕ ∕ ∕ t(6 + t + t))6/9 it u0/9
1
(u
iu 1 u6/9 iu 1
iu 1 6)u 6)u6/9 iu 1
u0/9 iu
u(u6/9 )
u6/9 iu
u6/9 iu
u:/9 u0/9 u6/9 + 6 u0/9 + 6 n 1 + n 1 + n : 0 6 0 + 6 + 6 9 9 9 9 9 :/9 9 0/9 9 9 u + n 1 n 1 (6 + t 1 u + t)):/9 (6 + t + t))0/9 + n : 0 : 0
1
∕
64.
∕
(x5 + 6)∕>/5 ix
Sfrf `stf jbt`mrfk, bh `s b`n`sfrjh gfn`r ush i` kf sustjtunj¹ hb x 1 tfb 1 tfbηη
u 5 1 x5 + 6 , ku`mh, 5iu Yhafbih u Yhafbih 5iu 1 1 5xix
=
uiu 1 xix xix ↕ uiu 1
u 5 1 x 5 + 6, x sj sj u 6, x 5 1 u 5
∕ u5
∕ 6 , y, x y, x 1
sustjtuy`bih
6, phr tfbth, ix tfbth, ix 1 1
u iu iu 1 ix 1 (u ) (x + 6) u5 6 6 6 iu 1 thafbih kf sustjtunj¹hb u hb u 1 1 , iu 1 t t5
5
↕
∕>/5
iu u5 u5
∟ ∕ 6
5 ∕>/5
∟ ∕ ∕
6 it t5
∕ ↕
1
it
∕
∕>
u
1
∟ u5u∕
u iu iu 1 1 u5 6
∟ ∕
6
iu
iu u5 u5
tit 6 t5
∕ ∟
6∕t ∕ ∕ 6 t 6 s`f s s`f s 1 1 6 ∕ t5 , ku`mh, is ku`mh, is 1 1 ∕5tit ↕ tit tit 1 1 ∕ is 5 ∕∟ 65 is 1 6 ∟ is 6 tit ↕ ∕ ∕ ∟ s∕6/5 is 1 5 s 5 s 5 6∕t 6 t5
6 t5
5
5
∕ ∕ ∕ ∕ ∕ ∟ ∕6/5+6
6/5
n 1 56 s 6 1 65 s 6 + 6 + n 1 5 5 1 (6
5 6/5
t )
n 1 + n 1
6
u5
x 1 ^ustjtuy`bih x ^ustjtuy`bih
ix 1 (x5 + 6)∕>/5 ix 1
6=.
n 1 s s 6/5 + n + n 1
6/5
6 u5
n 1 + n 1
6 , y, u y, u 1 1
u5 6 u5
6/5
n 1 + n 1
x5 + 6 , s` hetj`b`?
x n 1 x x((x5 + 6)∕6/5 + n + n 1 5 x +6
x5 (7x> + 5=)5/> ix
u 1 7x> + 5= , iu , iu 1 59x 59x5 ix
5
>
5/>
x (7x (7x + 5=)
ix
↕
iu 1 x 5 ix ↕ 596 iu 1 x
6 59
5/>
u
6 u5/>+6 iu iu 1 + n 1 59 5> + 6
6 0/> 6 u0/> 6 n 1 u + n 1 n 1 + n 1 1 (7x (7x> + 5=)0/> + n 0 93 59 93 >
67.
(s`bx + nhsx)ix (s`bx nhsx)6/>
∕
u 1 s`bx thafbih u thafbih 1 s`bx
nhsx , , iu 1 iu 1 (s`bx s`bx + nhsx))ix ∕ nhsx + nhsx
7
∟ u5 ∕ u
6 + n
∟ ∕ 6
5
>
5/>
x (7x (7x + 5=)
ix
iu 1 u6/>
↕
u∕6/> iu
u5/> u∕6/>+6 > n 1 n 1 u5/> + n + n 1 + n 1 5 ∕>6 + 6 5 > > 1 (s`bx nhsx nhsx))5/> + n 5
1
∕
6:.
xix
6 + x5
(6 + x5 )>
u 1 6 + x + x5 , iu 1 5xix 6 5
iu
6 1 5 u>
iu
∟ u
iu
6 1 5 u + u + u u 6 u , it 1 iu 5 u
∟ u + ∟ t 1 thafbih t thafbih 1 6 5
↕ 56 iu iu 1 xix 1 xix
∟ 6+ u
∟
53.
(x5 + 6 5x)6/0 ix 1 6 x
6 1 5 u)
∟ u(6 +
u 6+
∕ ∕
∟ u
6 + x + x5 + n
∕ 6)5/0ix 1 ∕ (x∕6)5/0∕6ix ix 1 1 ∕ ∕(x ∕ 6) (x ∕ 6)∕>/0+6 (x ∕ 6)5/0 0 n 1 n ∕ n 1 ∕ (x ∕ 6)5/0 + n 1∕ + 1 + n 1 > 5 5 ∕ 0 + 6 0
iu
∟
∕
u + n + n 1 1 5 6 +
(x5 + 6 5x)6/0 ix 6 x
∕ ∕
iu
t))∕6/5+6 it (6 + + t (6 + t + t))6/5 n n 1 1 + 1 + n 6 6 6 + t + t 5 + 6 5
↕ ∟ ∟
n 1 5 6 + 1 5(6 + t + t))6/5 + n 1
∟
(x
∕
(x 6)∕>/0 ix
hraukfs hraukfs i` khs t`hr`afs t`hr`afs 6.67 y 6.6: phr a`ijh i`k a¹ `thih `thih 56. I`iunjr kfs d¹ i` sustjtunj¹ hb. hb. ¹ sj HB. sj Y@HU@AF 6.67 JBTFUJFBNJF DU@BY@ F PBF YUF^KFNJHB. d d `s n s` tj`b`? `s jbt`mrfek` `b \f, \ f, eQ, pfrf nfif b¹ u ua`rh a`rh r`fk r`fk n e+n
e
d ( d (x)ix ix 1 1
f+n
f
o n , ix io , f y x 1 e s`f x 1 s`f x 1 o , ix 1 1 io , pfrf x pfrf x 1 1 f y x 1 e ku`mh
∕
:
d d ((x
∕ n)ix
e+ ↕ o 1 f f++ n y y o o 1 1 e + n r`sp`ntjvfa`bt`,
e+n
e
d ( d (x)ix ix 1 1
d d (( o
f+n
f
∕ n)io
Fpkjnfbih `k t`hr`af dubifa`btfk i`k n¹ffknukh knukh s` tj`b`? e
d ( d (x)ix ix 1 D ((e) 1 D
f
e+n
d ( o
ix 1 D ((e + n D ((f + n ∕ n)ix 1 D + n)) ∕ D + n))
d ( d (x
∕ n)ix ix 1 D ((e + n D ((f + n 1 D + n)) ∕ D + n))
f+n e+n
f+n e+n
e
ifih qu`
D ((f) ∕ D
d ( d ( o
d ( d (x)ix ix 1 1
f+n
f
∕ n)io , D ( D (e) ∕ D D ((f) 1 D D ((e + n D ((f + n + n)) ∕ D + n)) e+n
e
d d ((x
d ( d (x)ix ix , , y ,
kh nufk i`au`strf qu` kfs jbt`mrfk`s
f+n
f
n)ix ix shb shb jmufk`s.
∕
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e
f
6 d ( d (x)ix ix 1 1 l
le
d
lf
o 6 ix 1 thafbih x 1 , ix thafbih 1 io , pfrf pfrf x 1 f y x 1 e l l a`bt`, ku`mh e
f
6 d (x)ix ix 1 1 l
le
o d l
lf
e
kh nufk i`au`strf qu` kfs jbt`mrfk`s
f
x ix l
↕ o 1 l lff y o 1 l lee r`sp`ntjvf-
ix ix 1 D ((le le)) 1 D
6 d ( d (x)ix ix , , y , l
55. ^`f? x
D ( D (x, f) 1
3
63
t p it (t5 + f5 )q
∕ D D ((lf lf))
le
x ix ix shb shb jmufk`s. l
d
lf
f 2 3 y p y q shb q shb `bt`rhs phsjtjvhs. ihbi` f ihbi` p+6 +6∕ ∕5q D ( D (x/f, 6). I`ahstrfr qu` D qu` D ((x, f) 1 f p
x r`sp`ntjvfa`bt` f x/f ff p u p u p p+6 p+6 iu iu 1 f iu 1 f (f5 u5 + f5 )q (f5 (u5 + 6))q 3
t 1 fu,, it it 1 f iu iu,, pfrf t x,, u u 1 Yhafbih t Yhafbih 1 fu 1 f pfrf t 1 1 3 , y, t y, t 1 1 x 1 3 , y, u y, u 1 1 x/f
D D ((x, f) 1
3
1
p+6 f p+6 f5q
x/f
3
x/f
fu) p (fu) iu 1 (f)iu 1 ((fu ((fu))5 + f5 )q
u p p+6 +6∕ ∕5q iu 1 iu 1 f f p (u5 + 6)q p
sj s` thaf m thaf m((t) 1
3
x/f
3
u p iu (u5 + 6)q
t , y, fpkjnfbih `k t`hr`af dubifa`btfk, (t5 + 6)q
x/f
m(t)it 1 it 1 M M((x/f x/f))
3
∕ M(3), i`k ajsah ahih, x/f
x/f
m(u)iu 1 M(x/f x/f))
3
m(u)iu iu,, shb jmufk`s, phr tfbth
3
p+6 p+6∕ ∕5q
D D ((x, f) 1 f
x/f
3
5>. I`ahstrfr qu`?
t p p+6 +6∕ ∕5q it 1 it 1 f f p D D ((x/f, 6), nhah s` qu`r¹ĶĶff prhefr. (t5 + 6)q
6
x
d ((t) 1 s`f d s`f
m(t)it, it, y,
M(3), kh nufk pru`ef qu` kfs jbt`mrfk`s
∕
3 x/f
it 1 6 + t + t5
6 6 , t it 1 1 , it 1 u 6 + t + t5
6/x
6
it x 2 3 sj sj x 3.. 6 + t + t5
∕ u65 iu
t 1 x , y, t 1 t 1 6, u 1 6 , y, u u 1 pfrf t pfrf 1 x 1 6 r`sp`ntjvfa`bt`, ku`mh x 6 u5 iu 6 6/x 6 + u5
6
6
6
d ( d (t) it 1 it 1
x
∕
1
∕
6/x
iu 1 6 + u + u5
6/x
6
iu 6 6 + u + u5
fpkjnfbih `k t`hr`af dubifa`btfk, 6/x
6
x
6
e
f
d (t)it 1 it 1 D D (6) (6)
∕ D ( D (x) <
d (x)ix 1 D (e) ∕ D (f) 1 ∕\D (f) ∕ D (e)Q 1 ∕
f
6
d d ((t)it 1 it 1 D D
∕ 6 x
d (x)ix, kh nufk pru`ef kf jmufkifi,
e
f
e
66
D D (6) (6) d (x)ix 1 ∕
f
e
d (x)ix
6/x
6
ifih qu`
d ( d (t)it 1 it 1
d ( d (u)iu iu , , `bthbn`s, D `bthbn`s, D (6) (6)
6
x
6
kh nufk pru`ef kf jmufkifi 59. I`ahstrfr qu`? 6
x
a
x (6
3
it 6 + t + t5 1
b
∕ x)
6/x
6 x
D D (6) (6)
it x 2 3 sj x 3.. 6 + t + t5 sj
6
6
ix ix 1 1
∕ D D ((x) 1 D
∕
xb (6
3
∕ x)a ix
a y b shb `bt`rhs phsjtjvhs. sj sj a thafbih u 1 6 x , thafbih u r`sp`ntjvfa`bt`
∕ ∕iu iu 1 ix , y, x 1 6 ∕ u. Sfrf x 1 ix, Sfrf x 1 3 y x 1 6 , u 1 6 y u 1 3
sustjtuy`bih? 3
6
a
x (6
3
d ((x) 1 x a (6 sj sj d hetj`b`
ix ix 1 1
∕
ub (6
6
∕ u)aib
∕ x)b , y, m (x) 1 xb(6 ∕ x)a, fpkjnfbih fpkjnfbih `k t`hr`af t`hr`af dubifa`btfk dubifa`btfk s` 6
6
∕
∕ x)
b
d d ((x) ix 1 D 1 D (6) (6)
3
∕ D (3)< D (3)<
3
m (u) iu iu 1 1
6
m(x) ix 1 M 1 M(6) (6)
3
∕ M(3), phr htrf pfrt`,
∕\M(3) ∕ M(6)Q 1 M 1 M(6) (6) ∕ M(3), nhb kh nufk s` pru`ef qu` 6
3
m(x) ix 1 3
phr tfbth,
6
6
6
m (u) iu
∕
xa (6
3
ix 1 ∕ x)b ix 1
xb (6
3
∕ x)a ix
50. I`ahstrfr qu`? σ/5 σ/5
a
∕a
a
nhs xs`b x ix ix 1 1 5
5 σ/ σ/5
3
3
65
nhsa x ix
a `s ub `bt`rh phsjtjvh. sj sj a σ/5 σ/ 5
a
σ/ σ/5 5
a
ix 1 nhsxs`bx)) ix 1 (nhsxs`bx
nhs xs`b x ix ix 1 1
5 σ/ σ/5
1
3
a
s`b5x s`b5 5
5s`bxnhsx
5
3
3
3
σ/ σ/5 5
a
5 σ/ σ/5
ix ix 1 1
3
s`ba 5x 6 ix ix 1 1 a a 5 5
5 σ/ σ/5
a
ix
a
s`b 5x ix 1 ix 1 5
∕a
5 σ/ σ/5
s`ba 5x ix
3
3
thafbih kf sustjtunj¹hb? hb? 5x 1 u 1 u + +
σ 6 , ix ix 1 iu,, y, gfnj`bih x 1 iu gfnj`bih x 1 1 3, u 1 5 5 5
∕a
σ/5 σ/5
a
s`b 5x ix
3
↕
s`b((u + fghrf,, t`bj`bih fghrf t`bj`bih `b nu`b nu`btf tf qu` qu` s`b 5∕a 5
σ/5 σ/ 5
σ σ ∕ σ5 , y, x x 1 1 , u 1 , ku`mh, 5 5 σ/ σ/5 5
5∕a 5
5 ∕σ/ σ/5
σ )iu 5
σ ) 1 nhsu 1 nhsu,, 5
σ 5∕a s`b (u + )iu iu 1 1 5 5 σ/5 ∕σ/5
s`ba (u +
a
5 σ/ σ/5
nhsa u iu
σ/5 5 ∕σ/
d pfr, jbt`mrfek` `b \3, efstf ahstrfr qu`, pfrf thif d pfr, \3, eQ s` sftjsdfn`? e
e
d (x) ix 1 5
d d ((x) ix
3
∕e
@b vjrtui i` kf fijtjvjifi r`sp`nth fk jbt`rvfkh i` jbt`mrfnj¹hb? hb? 3
e
d ( d (x) ix 1
e
d ( d (x) ix ix + +
d d ((x) ix
3
∕e
∕e
ifih qu` d qu` d `s `s jbt`mrfek` `b \ e, eQ y `b \3, \3, eQ, tfaej¹`b `b kh `s `b \ e, 3Q 3
∕e
∕e
d d ((x) ix 1
∕
∕
∕
d ( d (x) ix< ix < thafbih l thafbih l 1
3
∕6 `b `k t`hr`af i` ijkftfnj¹hb hb h nhb-
trfnnj¹ hb hb i`k jbt`rvfkh i` jbt`mrfnj¹hhb, b, s` tj`b`? ∕(∕e)
∕e
∕
d ( d (x) ix 1
3
∕(∕6)
e
d ( x)ix ix 1 1
∕
3
d `s d ( x) phr tfbth? nhah d nhah `s pfr, s` sftjsdfn` d sftjsdfn` d ((x) 1 d (
∕
e
e
3
d ( d ( x)ix ix 1 1
∕
3
6>
d (x)ix
3
d d (( x)ix
∕
ku`mh pfrf thif d thif d pfr s` sftjsdfn`, 3
e
5∕a 5
σ/5 σ/ 5
σ/5 σ/5
5∕a 5
nhs u iu iu 1 1 5
∕σ/5 σ/5
σ/ σ/5 5
nhah
a
tfaej¹``b b kh shb, shb , 5
∕a
nhs u iu iu 1 1 5
σ/5 σ/5
a
nhs u iu iu , , y, 5
∕a
5 σ/ σ/5
nhsa x ix ix,, ku`mh?
σ/ σ/5 5 a
a
nhs xs`b x ix ix 1 1 5
∕a
3
3
54. (f) I`ahstrfr qu`? σ xd ( xd (s`bx s`bx))ix ix 1 1 5
3
nhsa u iu
nhsa u iu shb jmufk`s, `b pfrtjnukfr,
σ/5 σ/5
σ
3
3
∕a
3
3
σ/5 σ/ 5
σ/ σ/5 5
a
3
nhsa xs`ba x ix , y , 5∕a
3
3
3
Y`bj`bih qu` nhs qu` nhs a x `s pfr 5 ,
d d ((x) ix ix..
3
3
∕e
∕e
d d ((x) ix ix 1 1 5
d (x) ix ix+ +
d ( d (x) ix ix 1 1
e
e
e
e
d ( d (x) ix+ ix +
d d ((x) ix ix 1 1
nhsa x ix >
σ
d ( d (s`bx s`bx))ix. \jbijnfnj¹hb? u σ hb? u 1 1 σ
3
∕ xQ
(e) Fpkjnfr (f) pfrf i`iunjr kf d¹hhraukf? raukf? σ
3
xs`bx ix ix 1 σ 1 σ 6 + nhs + nhs5 x
xd ((s`bx s`bx)) , y,u (f) s`f m s`f m((x) 1 xd y,u 1 σ 1 σ
6
3
ix . 6 + x + x5
∕ x, iu 1 iu 1 ∕ix
x 1 u 1 σ σ , y, u 1 3 r`sp`ntjvfa`bt` pfrf x pfrf 1 3 , y, x y, x 1 σ 1 σ , u 1 5
−
Kf pfrjifi i` nhsx s` pu`i` v`r dfnjka`bt` `b su i`sfrrhkkh i` s`rj` i` Yfykhr, nhsx 1
b13
x p hr ∕ hr ∕ x, −
nhs(∕x) 1 nhsx, nhs
>
a
5b
(∕ ∕x) b (
(∕6)
(5b)!
b13
a
x 1 \nhsxQ
−
1
b x
(∕6)
b13 a
1 \ nhs(∕x)Q
(5b)!
< sj s` sustjtuy`
5b
(5b)!
1 nhs
5b
b x
(∕6)
a
, kh nufk pru`ef qu` nhsx 1 nhs (∕x). Fghrf, `b vjrtui i` kf pfrjifi i`
(∕x), qu`ifbih i`ahstrfih qu` nhs a x `s pfr, pfrf a ub `bt`rh phsjtjvh.
bht`s` qu`, sj s` thaf, d (x) 1 nhs a xs`ba x , y , m (x) 1 5 a nhsa x , fpkjnfbih `k t`hr`af dubifa`btfk, s` hetj`b` ∕
σ/5
d (x) ix 1 D
3
ifih qu`,
σ/5
a
nhs
a
xs`b
σ
5
∕ D (3) <
m (x) ix 1 M
3
a
∕
x i x 1 5
σ/5
a
nhs
σ/5
3
nhs
a
xs`b
a
a
∕
x ix , y, 5
σ
5
∕ M(3)<
u iu , `bthbn`s, D
3
3
qu`, kfs jbt`mrfk`s
σ/5
σ/5
a
nhs
3
69
σ/5
m (u) i u 1 M
3
σ
5
∕ D (3) 1 M
x ix shb jmufk`s.
σ
5
σ
5
∕ M(3)
∕ M(3). I` f g¹ Ķ kf nhbnkusj¹ hb hb
3
σ
m(x)ix ix 1 1
3
∕
σ
(σ
d (s`b s`b((σ ∕ u)iu iu 1 ∕ u)d ( 1
σ
1 σ 1 σ
3 σ
d ( d (s`b s`b((σ u))iu ))iu
ud ( ud (s`b s`b((σ u))iu σ ))iu 1 1 σ
∕ ∕ ∕
∕
3
σ
m (u)iu
d ( d (s`bu s`bu))iu
(σ
3
s`b((σ ∕ u)) iu ∕ u)d (s`b ))iu σ
σ
σ
σ
d d ((s`bu s`bu))iu
3
∕
d ( d (s`bx s`bx))ix ix , , y,
m(x)ix ix 1 σ 1 σ
fs¹Ķ,
d (s`bu s`bu))iu
σ m(x)ix ix 1 1 5
σ
3
3
σ
m (x)ix ix 1 σ 1 σ
d d ((s`bu s`bu))iu
3
3
d ( d (s`bx s`bx))ix ix,, nhah s` qu`r¹ qu`r¹Ķf i`ahstrfr. i`ahstrfr .
3
x s`bx 6 d (s`bx s`bx)) 1 m (x) 1 , d ( , y, m( 5 5 x5 5 x 5 s`b x 6+ + x
∕
xs`bx ix ix 1 1 6 + nhs + nhs5 x
σ
3
∕
σ xd (s`bx s`bx))ix ix 1 1 5
σ
xs`bx ix ix 1 1 6 + (6 s`b5 x)
∕
σ
xd ( xd (s`bx s`bx))ix ix 1 1
3
σ
5
σ
i` kf pfrt` (f) s` tj`b`,
d (s`bu s`bu))iu iu,, ku`mh
d ( d (s`bu s`bu))iu iu,, phr tfbth
xd ( xd (s`bx s`bx))ix ix 1 1 σ 5
(e) s`f d s`f d ((x) 1 σ
m(u)iu iu,,
3
3
σ
m(u)iu
3
3
3
m(x)ix ix , , y,
σ
σ
σ
σ
σ
3
3
↕
3
σ
σ
shb jmufk`s, i`k ajsah ahih,
ud ( ud (s`bu s`bu))iu9
σ
`k t`hr`af dubifa`btfk, p`rajt` `stfek`n`r qu` kfs jbt`mrfk`s j bt`mrfk`s
3
3
3
3
σ
∕
σ
σ 5
s`bx σ ix ix 1 1 5 s`b5 x 5
3 σ
xs`bx ix ix 1 1 5 s`b5 x
∕
σ
xd ( xd (s`bx s`bx))ix
3
d (s`bx s`bx))ix ix , , ku`mh,
3 σ
s`bx ix 6 + nhs + nhs5 x
∕ 3 nhsx,, iu 1 ∕ s`bxix s`bxix
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