Funciones Trascendentes - Logarítmica. SG
October 11, 2022 | Author: Anonymous | Category: N/A
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@ZGNLÞG DOCHQÍ[MLNH
Nuhgko jstuklhmos dh @ugnlþg Jxpogjgnlhd, nogndulmos quj6
= 2 79 2 ≩8
Jsto lmpdlnh quj js ugh `ugnlþg ugo h ugo (Alyjntlvh), tljgj jg nogsjnujgnlh ugh `ugnlþg lgvjrsh quj vljgj h sjr dh @ugnlþg Dochrítmlnh. Dh `ugnlþg dochrítmlnh js ugh `ugnlþg kj dh `ormh6 ^j kjgoth6 ^lcgl`lnhkos6
=doc ⇛ =
6ℝ6 ℝ+ ↑ ℝ
y = Yotjgnlh (Gömjro rjhd y posltlvo)
h = Ahsj kjd dochrltmo x = Dochrltmo (rjsudthko)
U js jd jxpogjg jxpogjgtj tj hd q quj uj sj jdjvh dh ahsj h phrh oatjgjr y.
Qjjmpdhzhmos dh phdharh jxpogjgtj por dh phdharh DOCHQL[MO y klrjmos quj6
=
2 sj djj6
— x js jd dochrltmo kj y jg ahsj h —
JD DOCHQL[MO kj ug kj ug gömjro jg ugh ahsj khkh, posltlvh y klstlgth kj dh uglkhk, js jd jxpogjgtj hd nuhd sj kjaj jdjvhr jdj vhr dh ahsj phrh oatjgjr klneo gömjro.
= ⟰ =doc > =< ⟰ 1=doc < ⟰ doc . GO jxlstj dochrltmo sl dh AH^J js dh ZGLKHK6
∑ ddoocc;::==9.;33818
1. ^l ug gömjro rjhd js mjgor quj 8, pjro quj mhyor quj njro, su dochrltmo js GJCH[LPO. Jfjmpdo6
doc9.:= 9.>9891 doc89999=doc89 = ;
doc9.8=doc89∑∑= 8 doc9.98=doc89 ∑= 1 doc9.998=doc89 ∑= > doc9.9998=doc89 = ;
Jtn.
Jtn.
>. Dochrltmos kj ahsj6 a = 89
YQOYLJKHKJ^ YHQ[LNZDHQJ^ (ahsj6
44
)
8. ^l ug gömjro rjhd js mhyor quj 8, su dochrltmo js GJCH[LPO. Jfjmpdo6
doc. 1=8
1. ^l ug gömjro rjhd js mjgor quj 8, pjro quj mhyor h njro, su dochrltmo js YO^L[LPO. Jfjmpdo6
doc. 9.:=8
YQOYLJKHKJ^ OYJQH[LPH^
8. Dochrltmo kj ug prokunto6
.. = = = . ∖ = .
. Dochrltmo kj ugh potjgnlh6
88.
81.
:. Dochrltmo kj dh ahsj6
Nhmalo kj ahsj6
;. Dochrltmo kj ugh rhíz6
. = ∝ =∝ . . = = = = = ⇑ =
8>.
0. Dochrltmo kj dh uglkhk6
8;.
3. Yotjgnlh dochrítmln dochrítmlnh6 h6
8:.
== = = = ∖ OA^JQPHNLÞG6 ( ∖ )
80.
5.
83.
≩
Kj`lglnlþg Dochrítmlnh6
=
NODOCHQL[MO Jd nodochrltmo kj ug gömjro js lcuhd hd dochrltmo kj dh lgvjrsh kjd gömjro.
=
Hpdlnhgko propljkhkjs opjrhtlvhs, 1 y 0, tjgjmos6
= = ⇑ = = = = ∑ =∑ = = = = = =
Jfjmpdos6 h)
a)
Yrop. 8:6
a)
Yrop. 8:6
Yrop. 886
Yrop. :6
HG[LDOCHQL[MO
Js dh opjrhnlþg nogtrhrlh h dh dochrltmhnlþg, nuhgko hntöh soarj ug dochrltmo do jdlmlgh sljmprj quj jstìg jg dh mlsmh ahsj. d doc doc hdgömjro = norrjspogkljgtj þ doc dhddodochrltmo c = khko, jsto js, ^j kjgomlgh hgtldochrltmo
sl x js jd dochrltmo kj G jg ahsj a, jgtognjs6
Jfjmpdos6 h)
a)
d doc = ⇑ == = =
= =
J F J Q N L N LO LO ^ Q J ^ ZJ D [ O^ O^ K J J N ZH ZH N L OG OG J ^ D OC OC H Q Í [M [ML N H ^
8. Jxprjsj jg `ormh jxpogjgnlhd6 h) a)
= =
∑ ⇑⇑ +==
1. Jxprjsj jg ``ormh ormh dochrítmlnh6 h) a)
∑ = ⇑ = ,, = ⇑ , =
Nhdnudhr dos slculjgtjs gömjros6
a) = = ?? ∑ = ??? ? Yrop. 36 = .6 ∑ = =??? = n + =? k) + = . (Yrop. 8) 6 = =2 (^lstjmh kj dochrltmo gjpjrlhgo y prop.8:)
h)
)
+ = . = =,
⇑ 66 è
Oatjgnlþg kjd gömjro —j” jg dh nhdnudhkorh6 \ dj shdkrè nomo rjspujsth6 1.385158515….
=?=??? = 2 . 6 = = 2 .. j)
= =
Nhdnudhr jd vhdor kj dh slculjgtj jxprjslþg6 (^lmpdl`lnhr)
∖ (.(.∖ ∖ ) )∖ (.(.∖ ∖ )) ∑ . . = . = = = ∖ ∖ = ∖ = ∖ ∖ ∖ ∖
2
=
(Qhínjs jg potjgnlhs)
2
^odunlþg6
(Yrop. 8: y prop. jxpogjgtjs)
(Opjrhnlogjs y rhnloghdlzhnlþg)
Jxprjshr jd dochrltmo khko jg tìrmlgos kj dochrltmos doch rltmos mès sjgnlddos6
.
. 2 . .. . 2 R .]2 . R ]2 .
2 .
þ6
Jsnrlah dh jxprjslþg nomo ug sodo dochrltmo6
2 . ∖ ∖ ) . .( ( þ6 .. 2
Qjsodvjr dhs slculjgtjs jnuhnlogjs dochrítmlnhs6
+ = ∑
(Nuhgko dhs ahsjs sog kl`jrjgtjs sj tomh dochrltmo jg hmaos mljmaros)
+ =∑ = = 2 .
=2 =2 . ì . . = 2 ö . = 2 þ6 =,
+ =
∑ =2 þ í ∑ 2 ..∑ = ∑ ö ∑ . ∑ = 2 ∑ Phmos ehnjr ug nhmalo kj vhrlhadj6
∑ =
=
=2 = = ∑ = ∑ ∑ = 2 .= = 2 þ6 =
Qjjmpdhzhmos jd vhdor jgnogtrhko jg jd nhmalo kj vhrlhadj quj elnlmos6
=
∖ ∖ = 2 . = 2 .. ∖ = ∖
=.(∖ =.(∖ )2 )2 =. (∖ ) ) 2 =. =. 2 =2=2 þ6 = 2 =
YQOADJMH6 Zg mjklnhmjgto sj jdlmlgh kjd nujrpo h trhvìs kj dh orlgh. Dh kosls lglnlhd js kj 89mc y dh nhgtlkhk H(t) quj qujkh jg jd nujrpo t eorhs kjspuìs kjs puìs kj lgcjrlrdo jstè khko por6
= . ..
Yhrh quj jd `èrmhno ehch j`jnto kjaj ehajr jg jd nujrpo por do mjgos 1mc. Kjtjrmlghr6 Nuègko qujkhg sodo 1 mc. KH[O^6 H (t) = Nhgtlkhk kj mjklnhmjgto jg mldlcrhmos quj qujkh jg jd nujrpo t eorhs kjspuìs kj sumlglstrhko t =[ljmpo Jgtognjs6 H (t) = 1mc (Nhgtlkhk kj mjklnhmjgto pjrmltlko) t=? (tljmpo) H (t) = 89. Yhso 86 ^ustltulmos6
,, ..
1 = 89.
Yhso 16 Kjspjfhmos klvlkljgko jgtrj 89 y slmpdl`lnhmos, rjsudth6 rjsudth6
9,1 =
..
Yhso >6 Hpdlnhmos dochrltmo nomög jg hmaos mljmaros.
doc 9,1= doc
..
Yhso ;6 Hpdlnhmos dh propljkhk kj dos dochrltmos doc 9,1 = t doc 9,5 Yhso :6 Kjspjfhmos t6
t = 3,18 e
^odunlþg6 Yor do thgto thgto hproxlmhkhmjgtj 3 eorhs kjspuìs kj hkmlglstrhko jd mjklnhmjgto tjgkrè jg su nujrpo 1 mc
Jfjrnlnlos6 8.
doc (1) +doc(x+>) = doc(x+:) ^odunlþg
Dh sumh kj kos dochrltmos js jd dochrltmo kjd prokunto6
⋊
Doc (1 (x+>)) (x+>)) = doc(x+:) Doc (1x+0) = doc(x+:) Nomo dos kos dochrltmos sog lcuhdjs, sus hrcumjgtos tljgjg quj sjr lcuhdjs. Yor thgto, 1x+0 = x+:
Qjsodvjmos dh jnuhnlþg dh jnuhnlþg kj prlmjr crhko6
1x ∑ x = :∑0 U = ∑8 1.
8
Hpdlnhmos dh propljkhk kjd dochrltmo kj ugh potjgnlh jg dos kos mljmaros
1 Qjhdlzhmos
> Opjrhmos
dh propljkhk kjd dochrltmo kj ug prokunto
jg jd prlmjr mljmaro
; Hpdlnhmos
dh lg`jntlvlkhk kj dos dochrltmos phrh qulthr dochrltmos
: Qjsodvjmos
dh jnuhnlþg
0
Gl gl sog sodunlogjs porquj sl dos sustltulmos jg dh jnuhnlþg gos jgnogtrhmos nog dochrltmo 9 y dochrltmo kj ug gömjro gjchtlvo y thdjs dochrltmos go jxlstjg, por do quj dh öglnh sodunlþg js
^odunlþg6
>.
8 Mudtlpdlnhmos
jg dos kos mljmaros por
y do phshmos toko hd prlmjr
mljmaro
1 Nogslkjrhgko
> Qjhdlzhmos
> Qjsodvjmos
quj
y qulthgko kjgomlghkorjs6
ug nhmalo kj vhrlhadj
dh jnuhnlþg
; Kjsehnjmos
jd nhmalo kj vhrlhadj
YQOADJMH6 Dh vlkh mjklh kjd jstrognlo )
66 = ,∻=, è .í
^odunlþg6
Kj mhgjrh quj dh jxprjslþg kj ddhh mhsh kjd jstrognlo
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