Apuntes de Matemáticas para la Economía I

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]klúk bm Mzr}blckrmz bm Jlmkjlcz Mjhkúeljcz q CMJ]J:E

CP]KRMZECRMEÃRLJCZ PC_C AC

MJHKHEÅC L



Rmec 3

N}kjlhkmz bm crlcoam _mca

3+3+ Ac _mjc _mca

Ahz k²}emrhz rmcamz zm p}mbmk hrbmkcr jheh ahz p}khz bm }kc rmjc+

Ahz mkmrhz phzll~hz v3! ?! :! 2! + + +u s}m z}rimk ca jhkcr! zm aaceck k²}emrhz kc}rcamz$ma jmrh zm p}mbm jhkzlbmrcr h kh jheh }k k²}emrh kc}rca! khzhrhz kh ah dcrmehz,+

Acz hpmrcjlhkmz crle²mljcz bm cbljl²hk | e}alpaljcjl²hk zm p}mbmk dcjmr bmkrh bmahz k²}emrhz kc}rcamz pmrh ac rmzc | ac bl~lzl²hk khz aam~c c lkrhb}jlr ahz zli}lmkmzk²}emrhz7

jmrh7 : ∟ : 5 0!

kmicl~hz7 ? ∟ > 5 ∟2!

nrcjjlhkmz7 : ù = 5:

=+

Phr ckh! phbmehz jaczlffijcr ahz k²}emrhz bm ac eckmrc zli}lmkm7

Kc}rcamz7 K 5 v3! ?! :! 2! + + +uMkmrhz7 W 5 v+ + + ! ∟:! ∟?! ∟3! 0! 3! ?! :! 2! + + +uma jmrh g}kh jhk ahz mkmrhz phzll~hz | kmicl~hz

_cjlhkcamz7 S 5

e

k7 e! k ∄ W! k 5 0

Lrrcjlhkcamz7 zhk ahz k²}emrhz rmcamz s}m kh zhk rcjlhkcamz! jheh phr mgmepah∗?! π! m

_mcamz7 _ 5 Lrrcjlhkcamz & _cjlhkcamz



? N}kjlhkmz bm crlcoam _mca

Rhbhz mzhz k²}emrhz cpcrmjmk jheh zha}jl²hk bm mj}cjlhkmz7t ∟ 3 5 0 ↟ t 5 3!

t & : 5 0 ↟ t 5 ∟:!?t & 3 5 0 ↟ t 5 ∟3

?!

t? 5 ? ↟ t 5 ³∗?!

t? & 3 5 0 ↟ t 5 ³∗∟3 (∄ _ ↟ Zhk ahz k²}emrhz Jhepamghz7J 5 vc & ol 7 c! o ∄ _! l 5

∗∟3u+

E²MRHBHZ BM _CWHKCELMKRH ECRME ²CRLJH

Ac bmffikljl²hk bm ahz k²}emrhz kc}rcamz mz ac zli}lmkm7

3 ∄ K+ Zl k ∄ K! ceol²mk pmrmkmjm k & 3

^mcehz j²heh n}kjlhkc mzc bmffikljl²hk7: ∄ K ⇟ : & 3 5 2 ∄ K!:(? (∄ K ⇟ :(? & 3 5 =(? (∄ K+

Mzc bmffikljl²hk bm ahz k²}emrhz kc}rcamz khz lkrhb}jm ma prhjmzh bm lkb}jjl²hk7

Bmehzrcjl²hk phr lkb}jjl²hk

Mz }kc ²mjkljc pcrc prhocr }kc cffirecjl²hk rmcalwcbc zhorm jcbc k²}emrh kc}rca7

Ac cffirecjl²hk mz ~mrbcbmrc pcrc k 5 3

Zl ac cffirecjl²hk mz ~mrbcbmrc pcrc k ∄ K! mkhkjmz ceol²mk mz ~mrbcbmrc pcrcz} z}jmzhr7 k & 3 ∄ K

Mzh lepaljc s}m ac cffirecjl²hk mz jlmrc pcrc hbh k²}emrh kc}rca k+

Jheh mgmepah! phbmehz prhocr phr lkb}jjl²hk s}mkl53

l 5k$k & 3,

?+

Bmehzrcjl²hk Blrmjc

Ac lbmc mz prhocr ac cffirecjl²hk blrmjcemkm

Phr mgmepah! pcrc bmehzrcr s}m $c ∟ o,$c & o, 5 c? ∟ o?!z²hah mkmehz s}m hpmrcr blrmjcemkm $c ∟ o,$c & o, 5 c? ∟ oc & co ∟ o? 5 c? ∟ o?+



3+3+ Ac _mjc _mca :

Bmehzrcjl²hk phr jhkrcbljjl²hk h _mb}jlh cb coz}rb}e

S}mrmehz prhocr }kc dlp²hmzlz

Cz}elehz ah hp}mzh c ac dlp²hmzlz |

Rmrelkcehz jhk }kc jhkrcbljjl²hk

Jhkja}lehz s}m ac dlp²hmzlz mz ~mrbcbmrc

Jheh mgmepah! ehzrcrmehz s}m∗

? mz lrrcjlhkca7∗? ∄ S ↟ ∗

? 5c

onrcjjl²hk lrrmb}jloam+

∗? 5

c

o ↟? 5

c?

o?

↟c? 5 ?o?

↟c? mz pcr

↟c pcr

↟c 5 ?r

↟c? 5 2r 5 ?o? ↟ o? pcr ↟ o pcr+

Zl c | o zhk pcrmz! mkhkjmzc

okh p}mbm zmr }kc nrcjjl²hk lrrmb}jloam ↟ jhkrcbljjl²hk#∗

? (∄ S+

BMZLI]CABCBMZ! ^CAH_ COZHA]RH

Prhplmbcbmz bm Hrbmk bm _c!o!j ∄ _

3+ Z²hah }kc bm acz zli}lmkmz cffirecjlhkmz zm ~mrlffijc7 c 1 o! c 5 o! c ; o

?+ Zl c 1 o | o 1 j! mkhkjmz c 1 j

:+ Zl c 1 o mkhkjmz c & j 1 o & j pcrc hbh k²}emrh rmca j

2+ Zl c 1 o | j ; 0 mkhkjmz cj 1 ojZl c 1 o | j 1 0 mkhkjmz cj ; oj

Prhplmbcb+ Mkrm bhz k²}emrhz rmcamz blzlkhz mtlzmk lkffiklhz k²}emrhz rcjlhkcamz mlrrcjlhkcamz+

Bmffikljl²hk 3+3+3 Ma ^cahr Cozha}h bm t ∄ _ mz7

qtq 5 t! zl t ≭ 0!

∟t! zl t 1 0+

Rmkmehz ac bmffikljl²hk camrkcl~c7 qtq 5∗

t?+

Ac lbmc imhe²mrljc bma ~cahr cozha}h mz ac bm blzckjlc+ Phr mgmepah! ¸s}²m p}khz mz²ckc blzckjlc : bma 08 Ac rmzp}mzc mz qtq 5 :! mz bmjlr t 5 ³:+

Bmffikljl²hk 3+3+? Zl t! | ∄ _ ac blzckjlc mkrm t m | mz

b$t! |, 5qt

∟|q+



2 N}kjlhkmz bm crlcoam _mca

Prhplmbcbmz bma ~cahr cozha}ht! | ∄ _

3+ qtq 5 0 ⇐⇟ t 5 0

?+ q ∟ tq 5 qtq:+ qt|q 5 qtqq|q2+ qt & |q ≢ qtq & q|q=+ qqtq ∟ q|q q ≢ qt ∟ |q

Lkmr~cahz mk _

Lkmr~cah colmrh7 $c! o, zhk hbhz ahz ~cahrmz bm t ca s}m c 1 t 1 o+

Lkmr~cah jmrrcbh7 Vc! oU zhk hbhz ahz ~cahrmz bm t ca s}m c ≢ t ≢ o+

Lkmr~cah zmel%colmrh7 Vc! o, zhk hbhz ahz ~cahrmz bm t ca s}m c ≢ t 1 o+

Lkmr~cah zmel%colmrh7 $c! oU zhk hbhz ahz ~cahrmz bm t ca s}m c 1 t ≢ o+

Lkmr~cah zmel%colmrh7 Vc! ∛, zhk hbhz ahz ~cahrmz bm t ca s}m c ≢ t 1 ∛+

Lkmr~cah colmrh7 $c! ∛, zhk hbhz ahz ~cahrmz bm t ca s}m c 1 t+

Lkmr~cah zmel%colmrh7 $∟∛! oU zhk hbhz ahz ~cahrmz bm t ca s}m t ≢ o+

Lkmr~cah colmrh7 $∟∛! o, zhk hbhz ahz ~cahrmz bm t ca s}m t 1 o+

Lkmr~cah colmrh7 $∟∛! ∛, zhk hbhz ahz ~cahrmz bm t ca s}m t ∄ _+

Khc+ ∛ kh mz }k k²}emrh rmca! lkbljc s}m ma lkmr~cah zm mtlmkbm zlk a²ĳelm+

Bmffikljl²hk 3+3+: Zmc C }k jhkg}kh bm k² }emrhz

Ma z}prmeh bm C mz ma emkhr mamemkh bm _ ec|hr h li}ca c hbhz ahz mamemkhzbm C+ Mz ac emkhr bm acz jhcz z}pmrlhrmz+

Ma ²ĳkffieh bm C mz ma ec|hr mamemkh _ emkhr h li}ca c hbhz ahz mamemkhz bmC+ Mz ac ec|hr bm acz jhcz lknmrlhrmz+

Zl ma z}prmeh pmrmkmjm c C mz ceol²mk ma e²ctleh bm C+

Zl ma ²ĳkffieh pmrmkmjm c C mz ceol²mk ma e²ĳkleh bm C+



3+?+ N}kjlhkmz mamemkcamz =

3+?+ N}kjlhkmz mamemkcamz

]kc n}kjl²hk n mk ahz rmcamz mz }kc rmiac s}m czlikc c jcbc k²}emrh t }k ²}kljhk²}emrh rmca n $t,7

t ↟ | 5 n $t,

t7 ~crlcoam lkbmpmkblmkm! cri}emkh h mkrcbc+|7 ~crlcoam bmpmkblmkm! ~cahr bm ac n}kjl²hk h zcalbc+

Bmffikljl²hk 3+?+3 Zmc B ⊏ _ }k jhkg}kh+ ]kc n}kjl²hk n jhk bhelklh B mz }kc rmiac s}m czlikc }k ² }kljh k² }emrh rmca n $t, c jcbc k² }emrh t mk B+B mz ma bhelklh bm n 7 Bhe$n , 5 B+

Ma jhkg}kh bm hbhz ahz ~cahrmz bm n nhreck ma rckih h ac lecimk bm n 7Le$n , 5 v| ∄ _ 7 ∊ t ∄ B! n $t, 5 |u+

Khcjl²hk7 n 7 B ⊏ _ ↟ _+

Pcrc ~lz}calwcr }kc n}kjl²hk mz e}| ²}la blo}gcr z} ir²cffijc7

Bmffikljl²hk 3+?+? Zmc n }kc n}kjl² hk jhk bhelklh B! ma jhkg}kh bm p}khz$t! n $t,, ∄ _? jhk t ∄ B nhreck ac ir²cffijc bm n +

¸J²heh phbmehz rmjhkhjmr ac ir²cffijc bm }kc n}kjl²hk8 Phbmehz blo}gcr a²ĳkmcz ~mrl%jcamz! }kc j}r~c s}m jhrc jcbc a²ĳkmc ~mrljca jheh e}jdh }kc ~mw! mz ac ir²cffijc bm }kcn}kjl²hk+

Phr mgmepah! |? 5 t lmkm bhz zha}jlhkmz | 5 ³∗t! phr ah s}m dc| bhz jhrmz jhk

jcbc a²ĳkmc ~mrljca pcrc t ; 0! phr ah ckh kh mz ac ir²cffijc bm }kc n}kjl²hk+

Zl blo}gcehz a²ĳkmcz ~mrljcamz! phbmehz hozmr~cr s}m ma bhelklh bm }kc n}kjl²hk mzma jhkg}kh bm t0 s}m ~mrlffijck s}m ac a²ĳkmc ~mrljca t 5 t0 jhrc ac ir²cffijc+ Bm ac elzecnhrec! zl blo}gcehz a²ĳkmcz dhrlwhkcamz! phbmehz ~lz}calwcr ac lecimk jheh ma jhkg}khbm |0 s}m ~mrlffijck s}m ac a²ĳkmc dhrlwhkca | 5 |0 jhrc ac ir²cffijc+

]kc n}kjl²hk zm p}mbm bmffiklr emblckm }kc n²hre}ac! }kc ir²cffijc h }kc bmzjrlpjl²hk+

Bmffikljl²hk 3+?+: Zmc n 7 B ↟ _ }kc n}kjl² hk7

3+ n mz }kc n}kjl² hk lk|mjl~c zl t3 5 t? lepaljc s}m n $t3, 5 n $t?,!h! bm eckmrc ms}l~camkm! zl n $t3, 5 n $t?, mkhkjmz t3 5 t?!mzh mz! zl czlikc cri}emkhz blzlkhz c ~cahrmz blzlkhz+

?+ n 7 B ↟ _ zm bljm zhorm|mjl~c zl Le$n , 5 _+n 7 B ↟ B zm bljm zhorm|mjl~c zl Le$n , 5 B+

:+ n mz }kc n}kjl²hk ol|mjl~c h }kh%c%}kh zl mz c ac ~mw lk|mjl~c | zhorm|mjl~c+



> N}kjlhkmz bm crlcoam _mca

3+:+ A²ĳelmz

Zmc n $t, }kc n}kjl²hk bmffiklbc pcrc hbh t jmrjc bm t0! pmrh kh kmjmzcrlcemkm mk maprhplh t 5 t0! zl ma ~cahr n $t, bm n zm cprhtlec ca k²}emrh A j}ckbh t zm cprhtlec cak²}emrh t0! bmjlehz s}m A mz ma a²ĳelm bm n $t, j}ckbh t lmkbm c t0+

^mcehz ma jhephrcelmkh bm ac zli}lmkm n}kjl²hk mk acz prhtlelbcbmz bm t 5 =7

n $t, 5?t? ∟ 9t & :

t ∟ :

t ?+< ?+<<<< ?+<<<<<< ↟ : ← :+000003 :+0003 :+3

n $t, 2+4 2+<<<4 2+<<<<<4 ↟ = ← =+00000? =+000? =+?a²ĳet↟:

n $t, 5 =

n $t, 5 ?t

?

∟ 9t & :t ∟ : 5 $?t ∟ 3,$t ∟ :,t ∟ : 5t5:

?t ∟ 3+

Bmffikljl²hk 3+:+3 $Ymlmrzrczz! bmffikljl²hk ∟ β, Zmc n }kc n}kjl²hk bmffiklbc mk }k lkmr~cah colmrh s}m jhklmkm c t0 $mtjmph phzloamemkm mk t0, | zmc A }k k² }emrhrmca! mkhkjmz

a²ĳet↟t0

n $t, 5 A!

zl ∀ ; 0! ∊β ; 0! 0 1 qt ∟ t0q 1 β ⇟ qn $t, ∟ Aq 1 ν+

Khc+ Phbmehz }lalwcr ac bmffikljl²hk ν∟

β pcrc prhocr s}m a²ĳet↟:

$?t∟

3, 5 =+

Khc+ ]kc rmiac imkmrca e}| ²}la mz mzjrlolr n $t, 5 A | mtprmzcrah mk ²mrelkhz bmt ∟ t0 hbh ah s}m phbcehz! emblckm ac li}cabcb t 5 $t ∟ t0, & t0+

Khc+ Mk ac bmffikljl²hk ν∟β! A | t0 zhk k²}emrhz ffiklhz+ Rmkmehz bmffikljlhkmz zlelacrmzpcrc t ↟ ³∛ | ceol²mk zl ma ~cahr bma a²ĳelm kh mz }k k²}emrh ffiklh+ Rceol²mk phbmehzbmffiklr a²ĳelmz acmrcamz7 phr ac bmrmjdc bm t07 a²ĳe

t↟t&

0

n $t, | phr ac lws}lmrbc bm t07

a²ĳet↟t

∟

0

n $t,+

Nhrecz Lkbmmrelkcbcz

∛ ∟ ∛!∛∛ ! ∛ ¹ 0!

0

0! ∛0! 3∛! 00+

C emkhz s}m aami}mehz c }kc lkbmmrelkcjl²hk! mkmehz acz zli}lmkmz prhplmbcbmz+



3+:+ A²ĳelmz 9

PrhplmbcbmzZ}phkicehz s}m a²ĳe

t↟t0

n $t, | a²ĳet↟t0

i$t, mtlzmk! pcrc t0 ffiklh h ³∛

3+ a²ĳet↟t0

$j3n $t, & j?i$t,, 5 j3 a²ĳet↟t0

n $t, & j? a²ĳet↟t0

i$t,! j3! j? ∄ _

?+ a²ĳet↟t0

$n $t,i$t,, 5 a²ĳet↟t0

n $t, a²ĳet↟t0

i$t,

:+ a²ĳet↟t0

3

n $t,

5

3

a²ĳet↟t0

n $t,! zl a²ĳe

t↟t0

n $t, 5 0

2+ _miac bma rmmepacwh7 zl n | i jhlkjlbmk pcrc hbh t jmrjc bm t0 $kh kmjmzcrlcemkmlkja}|mkbh c t0,! mkhkjmz a²ĳe

t↟t0

n $t, 5 a²ĳet↟t0

i$t,

=+ _miac bm ac n}kjl²hk jhep}mzc7 zl a²ĳet↟t0

n $t, 5 A | a²ĳet↟A

d$t, 5 d$A,! mkhkjmz

a²ĳet↟t0

d$n $t,, 5 d$ a²ĳet↟t0

n $t,,

A²ĳelmz o²czljhz+ Zmc t0 ffiklh

c, a²ĳet↟t0

j 5 j! a²ĳet↟³∛

j 5 j

o, a²ĳet↟t0

t 5 t0! a²ĳet↟³∛

t 5 ³∛! a²ĳet↟³∛

3

t5 0

j, a²ĳet↟t0

tk 5 tk0

! k∄K

b, a²ĳet↟t0

k

∗t 5 k

∗t0! $∀t0 mk z} bhelklh,

m, a²ĳet↟t0

n $t, 5 n $t0, ↟ n}kjlhkmz rlihkhe²mrljcz mk z} bhelklh

Amec 3+:+? $Amec bma Zckbyljd, Zmc L }k lkmr~cah ca s}m t0 ∄ L + Zmck n ! i| d n}kjlhkmz bmffiklbcz mk L ! mtjmph phzloamemkm mk ma prhplh t0+ Zl a²ĳe

t↟t0

i$t, 5

a²ĳet↟t0

d$t, 5 A |! ∀t ∄ L ! t 5 t0! d$t, ≢ n $t, ≢ i$t,+ Mkhkjmz

a²ĳet↟t0n $t, 5 A+

Khc+ Phbmehz }lalwcr ma amec bma Zckbyljd pcrc prhocr s}m a²ĳet↟0

zmk t

t5 3+



4 N}kjlhkmz bm crlcoam _mca

3+2+ Jhklk}lbcb

]kc n}kjl²hk jhklk}c mz }kc n}kjl²hk pcrc ac s}m! lk}ll~cemkm! pms}mkˑhz jceolhzmk ac mkrcbc prhb}jmk pms}mkˑhz jceolhz mk ac zcalbc

Bmffikljl²hk 3+2+3 Zmc n }kc n}kjl² hk bmffiklbc mk $t0 ∟ p!t0 & p,! p ; 0

n zm bljm jhkrlk}c mk t0 ⇐⇟ a²ĳet↟t0

n $t, 5 n $t0,

K²hmzm s}m ac n}kjl²hk bmom mzcr bmffiklbc mk t0 pcrc phbmr zmr jhklk}c mk mzm p}kh+ Zlac n}kjl²hk kh mz jhklk}c mk t0 bmjlehz mkhkjmz s}m mz blzjhklk}c mk bljdh p}kh7

  n $t, mz blzjhklk}c mk t0 zl

a²ĳet↟t0

n $t, kh mtlzm! h

ma a²ĳelm mtlzm pmrh kh mz li}ca c n $t0,

Bmffikljl²hk 3+2+? ]kc n}kjl² hk n $t, mz jhklk}c mk

_! zl mz jhklk}c mk jcbc p}kh

$c! o,! zl mz jhklk}c mk jcbc p}kh bma lkmr~cah

Vc! oU! zl mz jhklk}c mk $c! o, |

n $c, 5 a²ĳet↟c&

n $t,! n $o, 5 a²ĳet↟o∟

n $t,

Prhplmbcbmz o²czljczZmck n! i jhklk}cz mk t0! mkhkjmz acz zli}lmkmz n}kjlhkmz zhk jhklk}cz mk t0

3+ j3n & j?i! j3! j? ∄ _?+ n i

:+3

n ! zl n $t0, 5 0

2+ N}kjl²hk jhep}mzc7 zl i mz jhklk}c mk t0 | n mz jhklk}c mk i$t0,! mkhkjmz n ◮imz jhklk}c mk t0

Cai}kcz n}kjlhkmz jhklk}czAcz zli}lmkmz n}kjlhkmz zhk jhklk}cz mk z}z bhelklhz

c, p$t, phalkhelhz!p$t,

s$t,n}kjlhkmz rcjlhkcamz! k

∗t

o, n}kjlhkmz rlihkhe²mrljcz7 zmk$t,! jhz$t,! ck$t,! crjzmk$t,! + + +

j, n}kjlhkmz dlpmro²haljcz7 zmkd$t,! jhzd$t,! + + +

b, mtp$t, | ak$t,



3+2+ Jhklk}lbcb <

Rmhrmec 3+2+: $Rmhrmec bma ^cahr Lkmremblh, Zl n $t, mz }kc n}kjl² hk jhklk}c mk Vc! oU | F mz j}cas}lmr k² }emrh mkrm n $c, | n $o,! mkhkjmz mtlzm }k ~cahr j ∄ Vc! oU

s}m ~mrlffijc n $j, 5 F +

Rmhrmec 3+2+2 $Rmhrmec bm Ohawckh, Zl n $t, mz }kc n}kjl² hk jhklk}c mk Vc! oU | n $c, ¹ n $o, 1 0! mkhkjmz mtlzm }k ~cahr j ∄ $c! o, s}m ~mrlffijc n $j, 5 0+

Bmffikljl²hk 3+2+= ]kc n}kjl² hk n zm bljm cjhrcbc zl ma jhkg}kh bm z}z ~cahrmz mz² c cjh%cbh+ Mzh mz! zl mtlzm }k k² }emrh E ; 0 ca s}m qn $t,q ≢ E! pcrc hbh t mk z} bhelklh+

Rmhrmec 3+2+> $Rmhrmec bma ^cahr Mtrmeh, Zl n $t, mz }kc n}kjl² hk jhklk}c mk ma lkmr~cah jmrrcbh Vc! oU! mkhkjmz n cajckwc z}z ~cahrmz e²ctleh | e²ĳkleh+ Mzbmjlr! mtlzmk }khz ~cahrmz te | tE mk Vc! oU ca s}m7

n $tE , ≭ n $t, ≭ n $te, pcrc hbh t ∄ Vc! oU+



Rmec ?

J²caj}ah Blnmrmkjlca mk }kc

~crlcoam

?+3+ Bmrl~cbcz

Ac bmrl~cbc khz prhphrjlhkc }kc eckmrc bm jcaj}acr ac czc bm jceolh bm }kc n}kjl²hk

Jcaj}acehz ac ~mahjlbcb emblc jheh ac rcw²hk mkrm ac blzckjlc rmjhrrlbc mk }k lkmr~cahbm lmeph d | ac ahkil}b bma lkmr~cah bm lmeph+

~emblc 5t$r & d, ∟ t$r,

d!

mzm ~cahr mz ac pmkblmkm e bm ac rmjc s}m pczc c rc~²mz bm ahz p}khz $r! t$r,, |$r & d! t$r & d,,+ Zl s}mrmehz jcaj}acr ac ~mahjlbcb mk }k lmeph r bmomehz hecr maa²ĳelm

~$r, 5 a²ĳed↟0

t$r & d, ∟ t$r,

d5

b

brt$r,

Bmffikljl²hk ?+3+3 ]kc n}kjl² hk n zm bljm bmrl~coam mk t ⇐⇟

a²ĳed↟0

n $t & d, ∟ n $t,

d

mtlzm | mz }k k² }emrh ffiklh+

Zl n mz bmrl~coam! mkhkjmz n $t, 5b

btn $t, 5 a²ĳe

d↟0

n $t & d, ∟ n $t,

dmz ac bmrl~cbc bm

n mk t+

Khc+ Ac n}kjl²hk n mtlzm mk ahz p}khz bma bhelklh bm n ca s}m ma a²ĳelm mtlzm | mzffiklh+

Bmffikljl²hk ?+3+? $Bmn+ camrkcl~c,

n $t0, 5 a²ĳet↟t0

n $t, ∟ n $t0,

t∟

t0



? J²caj}ah Blnmrmkjlca mk }kc ~crlcoam

_mjc ckimkm

Ac rmjc s}m pczc phr $t0! n $t0,, jhk pmkblmkm e 5 n

$t0, mz ac rmjc ckimkm c n $t,mk t07 | 5 e$t ∟ t0, & n $t0,+

Prhplmbcbmz

3+ $j3n & j?i, 5 j3n & j?i! j3! j? ∄ _+

?+ $n ¹ i, 5 n i & n i

:+n

i

5

n i ∟ n i

i?

Rmhrmec ?+3+: $_miac bm ac Jcbmkc, Zl i mz bmrl~coam mk t | n ah mz mk i$t,!mkhkjmz ac n}kjl² hk jhep}mzc n ◮ i mz bmrl~coam mk t! | ~mrlffijc

$n ◮ i,$t, 5 n $i$t,,i$t,

Khc+ Jhk ac rmiac bm ac jcbmkc phbmehz bmehzrcr s}m

$n ∟3,$t, 53

n $n ∟3$t,,+

Phbmehz }lalwcr mzc lbmklbcb pcrc jcaj}acr acz bmrl~cbcz $crjck t,

| $ak t,

+Bmrl~cbcz o²czljcz

3+ j 5 0

?+ $tk, 5 ktk∟3

:+ $mt, 5 mt! $ahi t, 53

t

2+ $zmk t, 5 jhz t! $jhz t, 5 ∟zmk t! $ck t, 53

jhz? t

=+ $crjck t, 5 33 & t?

! $crjzmk t, 5 3∗3 ∟ t?

>+ $t, 5 jhzd! $jhzd t, 5 zmkd t

Rmhrmec ?+3+2

n bmrl~coam ⇟ n jhklk}c

Rmhrmec ?+3+= $Rmhrmec bm _haam, Zmc n om bmrl~coam mk $c! o, | jhklk}c mk Vc! oU+Zl n $c, 5 n $o,! mkhkjmz mtlzm ca emkhz }k k² }emrh j ∄ $c! o, ca s}m

n $j, 5 0+



?+3+ Bmrl~cbcz :

Rmhrmec ?+3+> $Rmhrmec bma ^cahr Emblh, Zmc n om bmrl~coam mk $c! o, | jhklk}c mk Vc! oU! mkhkjmz mtlzm ca emkhz }k k² }emrh j ∄ $c! o, ca s}m

n $j, 5n $o, ∟ n $c,

o ∟ c!

h! ms}l~camkmemkm n $o, ∟ n $c, 5 n $j,$o ∟ c,+

Rmhrmec ?+3+9 $_miac bm A“Dhˎplca, Zmck n | i n}kjlhkmz bmrl~coamz mk $c! o,! mt%

jmph phzloamemkm mk ma p}kh t0 ∄ $c! o,+ Zl a²ĳet↟t0

n $t,

i$t,mz ac lkbmmrelkcjl²hk

0

0! mk%

hkjmz

a²ĳet↟t0

n $t,

i$t,5 a²ĳe

t↟t0

n $t,

i$t,!

zlmeprm s}m a²ĳet↟t0

n $t,

i$t,mtlzc h zmc lkffiklh+

Mtmkzlhkmz

Ac rmiac bm A“Dhˎplca zm p}mbm cpaljcr mk ahz jczhz zli}lmkmz7

Zl ac lkbmmrelkcjl²hk mz ∛

∛jhk hbhz ahz zlikhz phzloamz+

Zl ma a²ĳelm mz t0

↟ ³∛C ahz a²ĳelmz acmrcamz

Bmrl~cjl²hk lepa²ĳjlc

Zl ac mj}cjl²hk ~lmkm bcbc bm eckmrc lepa²ĳjlc! N $t! |, 5 0! dc| s}m bmrl~crac rmzpmjh

c t | c pcrlr bm cd²ĳ homkmrb|

bt+

Bmrl~cbcz bm hrbmk z}pmrlhr

Phbmehz dcaacr ac bmrl~cbc bm }kc bmrl~cbc7

b?n

bt?5 n $t,!

b:n

bt:5 n $t,! ¹ ¹ ¹ !

bkn

btk5 n $k,$t,



2 J²caj}ah Blnmrmkjlca mk }kc ~crlcoam

?+?+ Mtrmehz

Bmffikljl²hk ?+?+3 Zmc n }kc n}kjl² hk bmffiklbc mk }k lkmr~cah L 7

n $te, mz ma e²ĳkleh iahoca $h cozha}h, bm n mk L zl n $te, ≢ n $t,! ∀t ∄ L

n $tE , mz ma e²ctleh iahoca $h cozha}h, bm n mk L zl n $tE , ≭ n $t,! ∀t ∄ L

Khc+ _mjhrbmehz s}m zl n mz jhklk}c! mk }k lkmr~cah jmrrcbh | cjhcbh Vc! oU! acn}kjl²hk zlmeprm cajckwc z} e²ctleh | e²ĳkleh iahoca +

Bmffikljl²hk ?+?+? Zmc n }kc n}kjl² hk bmffiklbc mk }k lkmr~cah L ! zl mkmehz }k lkmr%~cah colmrh L 3 jhkmklmkbh c t0

n $t0, mz }k e²ĳkleh ahjca bm n mk L zl n $t0, ≢ n $t,! ∀t ∄ L 3

n $t0, mz }k e²ctleh ahjca $h rmacl~h, bm n mk L zl n $t0, ≭ n $t,! ∀t ∄ L 3

Bmffikljl²hk ?+?+: Zmc n }kc n}kjl² hk bmffiklbc mk t0+ n lmkm }k p}kh jr²ĳljh mk t0

zl

n $t0, 5 0 h n $t0, kh mtlzm

Rmhrmec ?+?+2 Zl n lmkm }k e² ctleh h e²ĳkleh ahjca mk t0! mkhkjmz t0 mz }k p}khjr²ĳljh bm n +

J²caj}ah bm mtrmehz iahocamz bm }kc n}kjl²hk mk }k lkmr~cah jmrrcbh Vc! oU

3+ Dcaac ahz p}khz jr²ĳljhz bm n mk $c! o,7 n $t0, 5 0 h n $t0, kh mtlzc

?+ M~ca²}c n mk jcbc p}kh jr²ĳljh bm $c! o,

:+ M~ca²}c n mk ahz mtrmehz bma lkmr~cah7 n $c, | n $o,

2+ Ma emkhr ~cahr mz ma e²ĳkleh iahoca | ma ec|hr! ma e²ctleh iahoca

Bmffikljl²hk ?+?+=

n mz }kc n}kjl²hk jrmjlmkm mk }k lkmr~cah L zl

∀t3! t?

∄L jhk t3 1 t? mkmehz

s}m n $t3, 1 n $t?,+

n mz }kc n}kjl²hk bmjrmjlmkm mk }k lkmr~cah L zl ∀t3! t? ∄ L jhk t3 1 t?

mkmehz s}m n $t3, ; n $t?,+

Rmhrmec ?+?+> Zmc n }kc n}kjl²hk jhklk}c mk }k lkmr~cah jmrrcbh Vc! oU | bmrl~coammk $c! o,

3+ Zl n $t, ; 0! ∀t ∄ $c! o, mkhkjmz n mz jrmjlmkm mk Vc! oU+

?+ Zl n $t, 1 0! ∀t ∄ $c! o, mkhkjmz n mz bmjrmjlmkm mk Vc! oU+

:+ Zl n $t, 5 0! ∀t ∄ $c! o, mkhkjmz n mz jhkzckm mk Vc! oU+



?+?+ Mtrmehz =

Jrlmrlh bm ac prlemrc bmrl~cbc

t 1 t0 t ; t0 t0 $p}kh jr²ĳljh,

n $t,

∟ & e²ĳkleh ahjca& ∟ e²ctleh ahjca∟ ∟ kh mtrmeh ahjca& & kh mtrmeh ahjca

Bmffikljl²hk ?+?+9 Zmc n bmrl~coam mk }k lkmr~cah colmrh L + Ac ir² cffijc bm n mz

jhk~mtc $j² hkjc~c dcjlc crrloc, mk L zl n mz jrmjlmkm+

j²hkjc~c$j² hkjc~c dcjlc cocgh, mk L zl n mz bmjrmjlmkm+

Rmhrmec ?+?+4 Zmc n }kc n}kjl² hk bmrl~coam bhz ~mjmz mk }k lkmr~cah colmrh L

Zl n $t, ; 0! ∀t ∄ L ! mkhkjmz ac ir² cffijc bm n mz jhk~mtc mk L +

Zl n $t, 1 0! ∀t ∄ L ! mkhkjmz ac ir² cffijc bm n mz j² hkjc~c mk L +

Bmffikljl²hk ?+?+< Zmc n }kc n}kjl² hk jhklk}c mk }k lkmr~cah colmrh L | zmc t0 ∄ L +

n lmkm }k p}kh bm lkmtl²hk mk t0 zl ac jhkjc~lbcb jceolc mk t0 $jhk~mtc →j² hkjc~c,+

Rmhrmec ?+?+30 Zl t0 mz }k p}kh bm lkmtl² hk bm n ! mkhkjmz n

$t0, 5 0 h n

$t0,kh mtlzm+

Rmhrmec ?+?+33 Zmc n }kc n}kjl²hk ca s}m n $t0, 5 0 | bmrl~coam bhz ~mjmz mk }k lkmr~cah colmrh jhkmklmkbh c t0

zl n $t0, ; 0! mkhkjmz n lmkm }k e²ĳkleh ahjca mk t0+

zl n $t0, 1 0! mkhkjmz n lmkm }k e² ctleh ahjca mk t0+

Zl n $t0, 5 0 ma jrlmrlh kh n}kjlhkc! p}mbm zmr j}cas}lmr jhzc+

n $t0, n $t0, ir²cffijc

& ∟ jrmjlmkm! j²hkjc~c∟ ∟ bmjrmjlmkm! j²hkjc~c& & jrmjlmkm! jhk~mtc∟ & bmjrmjlmkm! jhk~mtc

0 & e²ĳkleh ahjca0 ∟ e²ctleh ahjca

0 0 8



> J²caj}ah Blnmrmkjlca mk }kc ~crlcoam

?+:+ Ir²cffijcz

3+ Bhelklh?+ Lkmrzmjjl²hk jhk ma mgm t ↟ n $t, 5 0

Lkmrzmjjl²hk jhk ma mgm | ↟ n $0, 5 |

:+ Zlemr²ĳcz

n $∟t, 5 &n $t, ↟ pcrn $∟t, 5 ∟n $t, ↟ lepcr

Pmrlhbljlbcb ↟ n $t & R , 5 n $t,

2+ Cz²ĳkhcz7

^mrljca

↟a²ĳet↟t0

n $t, 5

³∛Dhrlwhkca ↟ a²ĳet↟³∛

n $t, 5 D

Hoalj}c ↟ a²ĳet↟³∛

n $t, ∟ $et & o, 5 0 ↟ e 5 a²ĳet↟∛

n $t,

t! o 5 a²ĳe

t↟∛$n $t, ∟ et,

=+ Jhklk}lbcb7 a²ĳet↟t0

n $t, 5 n $t0,

>+ Bmrl~cbc7 ehkhhk²ĳc | p}khz jr²ĳljhz

n $t, ; 0↟ jrmjlmkmn $t, 1 0 ↟ bmjrmjlmkmn $t, 5 0 h n $t, kh mtlzc ↟ p}khz jr²ĳljhz

9+ E²ctlehz | e²ĳklehz ahjcamz7 t0 ↟ p}kh jr²ĳljh

n $t0, 5 0! n $t0, ; 0 ↟ e²ĳkleh ahjcan $t0, 5 0! n $t0, 1 0 ↟ e²ctleh ahjca

n $t, 7 ∟ ↟ & ↟ e²ĳkleh ahjca

n $t, 7 & ↟ ∟ ↟ e²ctleh ahjca

4+ Jhkjc~lbcb

n $t, ; 0 ↟ jhk~mtcn $t, 1 0 ↟ j²hkjc~c

<+ P}khz bm lkmtl²hk+ Ac jhkjc~lbcb jceolc+ n $t0, 5 0 h n $t0,

30+ E²ctlehz | e²ĳklehz iahocamz



?+2+ Phalkhelh bm Rc|ahr 9

?+2+ Phalkhelh bm Rc|ahr

Ac lbmc mz cprhtlecr }kc n}kjl²hk n $t, phr }k phalkhelh P $t,+ Ma phalkhelh bmRc|ahr mz ma phalkhelh s}m emghr cprhtlec c }kc n}kjl²hk mk }k p}kh t0

Zl cprhtlecehz n $t, phr }kc jhkzckm ↟ P $t, 5 n $t0,

}kc rmjc ↟ P $t, 5 n $t0, & n $t0,$t ∟ t0,

Bmffikljl²hk ?+2+3 Zl n mz bmrl~coam k ~mjmz mk t0! mkhkjmz ma phalkhelh

P k$t, 5 n $t0, & n $t0,$t ∟ t0, &n $t0,

?#$t ∟ t0,? & ¹ ¹ ¹ n $k,$t0,

k#$t ∟ t0,k

mz ma phalkhelh bm Rc|ahr bm ircbh k bm n jmkrcbh mk t0+

Khc+ Pcrc t0 5 0 ma phalkhelh ceol²mk zm aacec Phalkhelh bm Ecj Ac}rlk+

M__H_

Ma phalkhelh mz }kc cprhtlecjl²hk bm n $t,! phr ah s}m zm jhemm }k mrrhrq_k$t,q 5 qn $t, ∟ P $t,q+ Mtlzmk e}jdcz n²hre}acz pcrc mzm mrrhr! pmrh ac lbmc mz s}mhbcz maacz ~mrlffijck

a²ĳet↟t0

n $t, ∟ P k$t,

$t ∟ t0,k5 0+

↟ _k$t, 5 h

$t ∟ t0,k

+ Khcjl²hk7 n $t, 5 h

i$t,

j}ckbh t ↟ t0 ⇐⇟

a²ĳet↟t0

n $t,

i$t,5 0+

Mk ma zli}lmkm mhrmec mkmehz }kc n²hre}ac pcrc ma mrrhr q_k$t,q7

Rmhrmec ?+2+? Zmc n $t, }kc n}kjl² hk bmrl~coam k & 3 ~mjmz mk }k lkmr~cah colmrh L !mkhkjmz ∀t0! t ∄ L mkmehz s}m

n $t, 5 P k$t, & _k$t, 5 n $t0, & n $t0,$t ∟ t0, & ¹ ¹ ¹ n $k,$t0,

k#$t ∟ t0,k & _k$t,!

_k$t, 5 n $k&3,$λ,

$k & 3,#$t ∟ t0,$k&3,! λ mz }k p}kh mk ma lkmr~cah colmrh bmffiklbh phr t0 | t+



Rmec :

Z}jmzlhkmz | Zmrlmz

:+3+ Z}jmzlhkmz bm k²}emrhz rmcamz

Bmffikljl²hk :+3+3 ]kc z}jmzl²hk bm k²}emrhz rmcamz vcku mz }kc cpaljcjl² hk s}mczlikc c jcbc k ∄ K }k k² }emrh rmca7

ck 7 K↟ _

c3! c?! c: + + + zhk ahz ²mrelkhz bm ac z}jmzl²hk+

ck mz ma ²mrelkh imkmrca+Ac z}jmzl²hk ceol²mk p}mbm mepmwcr mk k 5 07 c0! c3! c? + + ++

Bmffikljl²hk :+3+? ]kc z}jmzl² hk vcku zm bljm jhk~mrimkm zl a²ĳek↟∛

ck 5 A! pcrc A

ffiklh+

Ma a²ĳelm bm }kc z}jmzl² hk vcku mz A zl pcrc hbh ; 0 ∊ K ∄ K ca s}m zl k ; K ⇟qck ∟ Aq 1 ν+ $Dc| }kc bmffikljl² hk camrkcl~c pcrc A lkffiklh+,

Zl ac z}jmzl²hk kh mz jhk~mrimkm! bmjlehz s}m mz bl~mrimkm+Acz prhplmbcbmz bm ahz a²ĳelmz bm z}jmzlhkmz zhk acz elzecz s}m acz prhplmbcbmz bm ahza²ĳelmz bm n}kjlhkmz+

Pcrc dcaacr ma a²ĳelm bm }kc z}jmzl²hk phbmehz }lalwcr cai}kcz ²mjkljcz jheh7

Ma jhkjmph bm a²ĳelm bm }kc n}kjl²hk7

Zmc n $t, }kc n}kjl²hk | vcku ac z}jmzl²hk n $k, 5 ck+

Zl a²ĳet↟∛

n $t, 5 A! mkhkjmz a²ĳek↟∛

ck 5 A

Phbmehz }lalwcr hbcz acz dmrrcelmkcz pcrc jcaj}acr ma a²ĳelm bm }kc n}kjl²hk!jheh! phr mgmepah! ac rmiac bm A“Dhˎplca+

Ma amec bma Zckbyljd pcrc z}jmzlhkmz7

Zl a²ĳek↟∛

ck 5 a²ĳek↟∛

ok $ffiklc h lkffiklc, | vjku ~mrlffijc s}m

ck ≢ jk ≢ ok! ∀ k ∄ K!

mkhkjmz a²ĳek↟∛

jk 5 a²ĳek↟∛

ck 5 a²ĳek↟∛

ok+



? Z}jmzlhkmz | Zmrlmz

Bmffikljl²hk :+3+: ]kc z}jmzl² hk vcku zm bljm7

3+ cjhcbc z}pmrlhremkm zl ∊

J ∄_ ca s}m ck

≢J +

?+ cjhcbc lknmrlhremkm zl ∊ J ∄ _ ca s}m ck ≭ J +

:+ cjhcbc zl mz cjhcbc z}pmrlhremkm m lknmrlhremkm$ ∊ J 3! J ? ∄ _! + s+ J 3 ≢ ck ≢ J ?,+

Bmffikljl²hk :+3+2 ]kc z}jmzl² hk vcku zm bljm7

3+ ehk² hhkc jrmjlmkm zl ck 1 ck&3 $kh bmjrmjlmkm zl ck ≢ ck&3,+

?+ ehk² hhkc bmjrmjlmkm zl ck ; ck&3 $kh jrmjlmkm zl ck ≭ ck&3,+

:+ ehk²hhkc! zl mz }kh bm ahz jczhz prm~lhz+

Rmhrmec :+3+=

vcku ehk² hhkc | cjhcbc ⇟ vcku jhk~mrimkm

Rmhrmec :+3+> $Jrlmrlh bm Zhaw, Zl acz z}jmzlhkmz vcku | voku ~mrlffijck }kh bmahz zli}lmkmz cpcrcbhz7

3+ voku mz ehk² hhkc jrmjlmkm jhk a²ĳek↟∛

ok 5 ∛!

?+ voku mz ehk² hhkc bmjrmjlmkm! jhk ok 5 0 pcrc jcbc k ∄ K |

a²ĳek↟∛ck 5 a²ĳek↟∛ok 5 0+

Zlmeprm s}m a²ĳek↟∛

ck&3 ∟ ck

ok&3 ∟ ok5 A! mtlzc pcrcr A ffiklh h lkffiklh! mkhkjmz

a²ĳek↟∛

ck

ok5 a²ĳe

k↟∛

ck&3 ∟ ck

ok&3 ∟ ok

Rmhrmec :+3+9 $N²hre}ac bm Zlralki,

a²ĳek↟∛

k#

kkm∟k∗

?πk5 3



:+?+ Zmrlmz bm k²}emrhz rmcamz :

:+?+ Zmrlmz bm k²}emrhz rmcamz

]kc zmrlm mz ac z}ec bm }kc z}jmzl²hk bm ²mrelkhz

Phr mgmepah ↟ ac z}ec imhe²mrljc7K k50

rk 5rK &3 ∟ 3

r ∟ 3+

Bmffikljl²hk :+?+3 Zmc vcku }kc z}jmzl² hk! }kc zmrlm$lkffiklc, mz ac z}ec bm hbhz z}z²mrelkhz7

∛k53

ck 5 c3 & c? & c: & c2 & ¹ ¹ ¹ +

Ac z}ec pcrjlca bm k ²mrelkhz mz Z k 5 c3 & c? & c: & ¹ ¹ ¹ ck+Zl ac z}jmzl² hk vZ ku bm z}ecz pcrjlcamz jhk~mrim ca a²ĳelm Z ! mkhkjmz bmjlehz s}m ac zmrlm∛

k53 ck jhk~mrim! | Z mz ac z}ec bm ac zmrlm7

Z 5 a²ĳek↟∛

Z k 5 c3 & c? & c: & c2 & ¹ ¹ ¹ +

Mk hrh jczh! bmjlehz s}m ac zmrlm bl~mrim+

Prhplmbcbmz

3+

ck |

ok jhk~ ⇟

$j3ck & j?ok, 5 j3

ck & j?

ok jhk~+

?+ a²ĳek↟∛

ck 5 0 ⇟

ck bl~+

:+

ck jhk~ ⇟ a²ĳek↟∛

ck 5 0+ $Pmrh a²ĳek↟∛

ck 5 0

ck jhk~,

Rmhrmec :+?+? Ac z}ec imhe²mrljc jhk~mrim zl 0 1 qrq 1 3! mk mzm jczh

∛k50

rk 53

3 ∟ r+

Rmhrmec :+?+: Ac zmrlm mamzj²hpljc $ ck 5 ok ∟ ok&3,

∛k53

$ok ∟ ok&3, 5 $o3 ∟ o?, & $o? ∟ o:, & $o: ∟ o2, & $o2 ∟ o=, & ¹ ¹ ¹

~mrlffijc s}m Z k 5 o3 ∟ ok&3+Mzc zmrlm jhk~mrim ⇐⇟ a²ĳe

k↟∛ok 1 ∛ |

Z 5 o3 ∟ a²ĳek↟∛

ok+

Rmhrmec :+?+2 Ac p%zmrlm $ p 5 3 mz ac zmrlm cre² hkljc,

∛k53

3

k p5

3

3 p&

3

? p&

3

: p&

3

2 p& ¹ ¹ ¹ +

3+ jhk~mrim zl p ; 3+

?+ bl~mrim zl 0 1 p

≢3+



2 Z}jmzlhkmz | Zmrlmz

J_LRM_LHZ BM JHK^M_IMKJLC PC_C ZM_LMZ

3+ Jrlmrlh bm jhepcrcjl²hk blrmjc7

vck

u|

vok

ubhz z}jmzlhkmz bm ²mrelkhz

phzll~hz

0 1 ck ≢ ok! ∀ k ∟↟

ok jhk~ ⇟

ck jhk~

ck bl~⇟

ok bl~

?+ Jrlmrlh bm jhepcrcjl²hk mk ma a²ĳelm7 vcku | voku bhz z}jmzlhkmz bm ²mrelkhzphzll~hz

a²ĳek↟∛

ck

ok5 A! A ffiklh | phzll~h

⇚ck |

ok lmkmk ma elzeh jhephrcelmkh

ceocz jhk~mrimk h ceocz bl~mrimk

:+ Jrlmrlh bm ac rc²ĳw7 vcku z}jmzl²hk bm ²mrelkhz phzll~hz

a²ĳek↟∛

k

∗ck 1 3 ⇟

ck jhk~

a²ĳek↟∛

k

∗ck ; 3 ⇟

ck bl~

a²ĳek↟∛

k∗ck 5 3 ma jrlmrlh kh jhkja}|m

2+ Jrlmrlh bma jhjlmkm7 vcku z}jmzl²hk bm ²mrelkhz phzll~hz

a²ĳek↟∛

ck&3

ck1 3 ⇟

ck jhk~

a²ĳek↟∛

ck&3

ck; 3 ⇟

ck bl~

a²ĳek↟∛

ck&3

ck5 3 ma jrlmrlh kh jhkja}|m

=+ Jrlmrlh bm Amloklw pcrc zmrlmz camrkcbcz7vcku z}jmzl²hk bm ²mrelkhz phzll~hz

Zl ck&3 ≢ ck | a²ĳek↟∛

ck 5 0

⇚Ac zmrlm camrkcbc

$∟3,kck jhk~mrim jhkbljlhkcaemkm

$∟3,k&3ck



:+:+ Zmrlmz bm phmkjlcz =

Bmffikljl²hk :+?+=

CJ+ ck mz cozha}cemkm jhk~mrimkm zl qck

qmz jhk~mrimkm+

JJ+ Zl

ck mz jhk~mrimkm pmrh qckq mz bl~mrimkm! mkhkjmz ac zmrlm mz jhkbl%

jlhkcaemkm jhk~mrimkm+

Jhk~mrimkjlc cozha}c 5⇟ Jhk~mrimkjlc jhkbljlhkca

Kh jhk~mrimkjlc jhkbljlhkca 5⇟ Kh jhk~mrimkjlc cozha}c

M__H_

J}ckbh cprhtlecehz ac z}ec bm }kc zmrlm camrkcbc phr z}z prlemrhz k ²mrelkhz!

mkhkjmz

Z 5 Z K & _K 5K k53

$∟3k,ck & _K ⇟ q_K q ≢ cK &3

Khc+ Phbmehz bmrl~cr h lkmircr }kc zmrlm lkffiklc pcrc homkmr hrc zmrlm+

:+:+ Zmrlmz bm phmkjlcz

Bmffikljl²hk :+:+3 ]kc zmrlm bm phmkjlcz jmkrcbc mk t0 mz }kc zmrlm lkffiklc bm ac nhrec

n $t, 5

∛k50

ck $t ∟ t0,k 5 c0 & c3$t ∟ t0, & c?$t ∟ t0,? & c:$t ∟ t0,: & ¹ ¹ ¹

Rmhrmec :+:+? $Jhk~mrimkjlc bm acz zmrlmz bm phmkjlcz,]kc zmrlm bm phmkjlcz mk t0 ~mrlffijc }kh | z² hah }kh bm ahz zli}lmkmz cpcrcbhz7

3+ Ac zmrlm jhk~mrim z² hah mk t0

?+ Mtlzm }k k² }emrh rmca σ ; 0 ca s}m ac zmrlm mz

cozha}cemkm jhk~mrimkm mk qt ∟ jq 1 σ

bl~mrimkm mk

qt

∟j

q; σ

:+ Ac zmrlm mz cozha}cemkm jhk~mrimkm pcrc hbh t ∄ _

Khc+ σ mz ma rcblh bm jhk~mrimkjlc bm ac zmrlm bm phmkjlcz+ σ 5 0! σ 1 ∛ ²h σ 5 ∛+Ma jhkg}kh bm hbhz ahz ~cahrmz bm t pcrc ahz j}camz ac zmrlm jhk~mrim mz ma lkmr~cah bmjhk~mrimkjlc bm ac zmrlm+ Ma rcblh zm p}mbm homkmr emblckm acz zli}lmkmz nhre}acz7

3

σ5 a²ĳe z}p

k↟∛

k

qckq

3

σ5 a²ĳe

k↟∛

ck&3

ck

! zl ma a²ĳelm mtlzm



> Z}jmzlhkmz | Zmrlmz

Rmhrmec :+:+: Zl ac zmrlm bm phmkjlcz n $t, 5∛

k50 ck $t ∟ t0,k lmkm rcblh bm jhk%~mrimkjlc σ ; 0! mkhkjmz n $t, mz jhklk}c! bmrl~coam m lkmircoam mk $t0 ∟ σ! t0 & σ,+

Ac bmrl~cbc | ac lkmirca zm jcaj}ack ²mrelkh c ²mrelkh+ Ceocz lmkmk ma elzeh rc%blh bm jhk~mrimkjlc s}m n + Ma lkmr~cah bm jhk~mrimkjlc p}mbm zmr blnmrmkm! bmolbh ca jhephrcelmkh mk ahz mtrmehz $ t 5 t0 ³ σ,+

Prhplmbcbmz+Zmck n $t, 5

∛

k50 cktk | i$t, 5∛

k50 oktk+

3+ n $ft, 5∛

k50 ckfktk

?+ n $tK , 5∛

k50 cktKk

:+ j3n $t, & j?i$t, 5∛

k50$j3ck & j?ok,tk

Bmffikljl²hk :+:+2 Zl mtlzmk hbcz acz bmrl~cbcz bm n mk t0! mkhkjmz ac zmrlm

∛k50

n $k,$t0,

k#$t ∟ t0,k

zm aacec ac zmrlm bm Rc|ahr bm n jmkrcbc mk t0 $pcrc t0 5 0 ceol²mk zm aacec ac zmrlm bm Ecj Ac}rlk bm n ,+

Rmhrmec :+:+= Zl mtlzmk hbcz acz bmrl~cbcz bm n mk }k lkmr~cah colmrh L s}m jhk%lmkm c t0 mkhkjmz

n $t, 5∛

k50

n $k,$t0,

k#

$t

∟t0,k

zl | zhah zl mtlzm λ mkrm t | t0 ca s}m

a²ĳek↟∛

_k$t, 5 a²ĳek↟∛

n $k&3,$λ,

$k & 3,#$t ∟ t0,k&3 5 0! ∀ t ∄ L+

ZM_LMZ BM RC\AH_

mt 5 3 & t &t?

?#&

t:

:#& ¹ ¹ ¹ &

tk

k#¹ ¹ ¹ ! ∟∛ 1 t 1 ∛

zmk t 5 t ∟ t:

:#& t

=

=#∟ t

9

9#& ¹ ¹ ¹ & $∟3,k t

?k&3

$?k & 3,#¹ ¹ ¹ ! ∟∛ 1 t 1 ∛

jhz t 5 3 ∟ t?

?#&

t2

2#∟ t>

>#& ¹ ¹ ¹ & $∟3,k

t?k

$?k,#¹ ¹ ¹ ! ∟∛ 1 t 1 ∛

crjck t 5 t ∟ t:

:&

t=

=∟ t9

9& ¹ ¹ ¹ & $∟3,k

t?k&3

$?k & 3,¹ ¹ ¹ ! ∟3 ≢ t ≢ 3

3

3 ∟ t5 3 & t & t? & t: & ¹ ¹ ¹ & tk ¹ ¹ ¹ ! ∟3 1 t 1 3

ak $3 & t, 5 t ∟ t?

?&

t:

:∟ ¹ ¹ ¹ & $∟3,k&3

tk

k¹ ¹ ¹ ! ∟3 1 t ≢ 3



Rmec 2

Lkmircjl²hk

2+3+ Prlell~cz

AC LKRMI_CA BMNLKLBC

Zl n $t, mz }kc n}kjl²hk jhklk}c | kh kmicl~c bmffiklbc mk ma lkmr~cah t ∄ Vc! oU! mkhkjmzac lkmirca bmffiklbc o

cn $t,

rmprmzmkc ma ²crmc ocgh ac ir²cffijc bm ac n}kjl²hk n $t, | zhorm ma mgm t mk ma lkmr~caht ∄ Vc! oU+

Zl n $t, mz }kc n}kjl²hk imkmrca! ac lkmirca bmffiklbc rmprmzmkc ac z}ec bm ²crmczjhk zlikh mkrm ac n}kjl²hk | ma mgm t+ Ac lbmc pcrc jcaj}acr ac lkmirca bmffiklbc mzbl~lblr ma lkmr~cah mk k z}olkmr~cahz | cprhtlecr ac n}kjl²hk phr }kc jhkzckm $maemkhr h ma ec|hr ~cahr! ma ~cahr bma p}kh emblh h j}cas}lmr ~cahr bm ac n}kjl²hk mk malkmr~cah,+ Mkhkjmz! jcaj}acehz ma ²crmc bm k rmj²cki}ahz! s}m mz e}jdh e²cz zmkjlaah+Zl cprhtlecehz n n $tl, mk ma lkmr~cah l%²mzleh! jhk hbhz ahz lkmr~cahz bm ac elzecahkil}b! ∎t! mkhkjmz phbmehz cprhtlecr ac lkmirca jheh

oc

n $t, n $t3,∎t & n $t?,∎t & ¹ ¹ ¹& n $tk,∎t+

Zl cprhtlecehz ac n}kjl²hk mk ma lkmr~cah phr ahz zli}lmkmz ~cahrmz! homkmehz ahz s}mzm aacec! pcrc bljdc pcrljl²hk bma lkmr~cah

Ma emkhr ~cahr mk jcbc z}olkmr~cah ↟ z}ec lknmrlhr bm n Ma ec|hr ~cahr mk jcbc z}olkmr~cah ↟ z}ec z}pmrlhr bm n J}cas}lmr ~cahr mk jcbc z}olkmr~cah ↟ z}ec bm _lmeckk bm n

Ca hecr ma a²ĳelm ∎t ↟ 0 zl hbcz acz z}ecz jhlkjlbmk! bmjlehz s}m ac n}kjl²hk mzlkmircoam _lmeckk+

Khc+ J}cas}lmr n}kjl²hk jhklk}c c rhwhz mz lkmircoam+

Rmkmehz ac zli}lmkm bmffikljl²hk bm lkmirca bmffiklbc7



? Lkmircjl²hk

Bmffikljl²hk 2+3+3 Bcbc }kc n}kjl²hk lkmircoam n $t, mk ma lkmr~cah Vc! oU! zm bl~lblehz

ma lkmr~cah mk k z}olkmr~cahz bm li}ca ahkil}b ∎t! malilmkbh j}cas}lmr p}kh tl mk

jcbc z}olkmr~cah! mkhkjmz zm bmffikm ac lkmirca bmffiklbc h ac lkmirca bm _lmeckk bmn $t, bm c c o jheh o

cn $t,bt 5 a²ĳe

k↟∛

kl53

n $tl ,∎t+

Khc+ Ahz e²mhbhz k}e²mrljhz zm oczck mk mzm jhkjmph bm jcaj}acr }kc lkmirca cprhtl%eckbh ma ~cahr bm ac n}kjl²hk mk jcbc z}olkmr~cah phr }kc jhkzckm h phr }k phalkhelh!e}jdh e²cz zmkjlaah bm lkmircr+

Prhplmbcbmz bm ac lkmirca

3+ oc j3n & j?i 5 j3

oc n & j?

oc i

?+

oc

n 5

jc

n &

oj

n

:+

oc

n 5 ∟ co

n

2+

cc

n 5 0

=+ oc n i 5 o

c n oc i

>+ n ≭ i ⇟ oc

n ≭ oc

i

9+ n ≭ 0 ⇟ oc

n ≭ 0

zl n ≢ 0 ⇟ oc

n ≢ 0

4+ oc

n ≢

ocqn q

<+ e ≢ n $t, ≢ E! ∀t ∄ Vc! oU ⇟e$o∟c, ≢ o

c n $t, ≢ E $o∟c,

AC LKRMI_CA LKBMNLKLBC

Imhe²mrljcemkm! ac bmrl~cbc z}rim ca dcaacr ac pmkblmkm bm }kc j}r~c | ac lkmircaca jcaj}acr ma ²crmc ocgh }kc j}r~c! pmrh Kmyhk bmzj}orl²h! cbme²cz! s}m bmrl~cr mlkmircr zhk prhjmzhz lk~mrzhz7

Bmrl~cjl²hk7 bcbc }kc n}kjl²hk N $t,! dcaac }kc n}kjl²hk n $t, s}m zclzncic

bN $t,bt

5 n $t,+

Lkmircjl²hk7 bcbc }kc n}kjl²hk n $t,! dcaac }kc n}kjl²hk N $t, s}m zclzncic

bN $t,

bt5 n $t,+

]kc n}kjl²hk N $t, s}m rmz}ma~m ma ²}aleh prhoamec zm aacec }kc prlell~c! cklbmrl~c%bc h lkmirca lkbmffiklbc bm n $t,+Ma prhoamec bm bmrl~cjl²hk zlmeprm lmkm zha}jl²hk pmrh ma bm lkmircjl²hk kh zlmeprm lmkmzha}jl²hk |! mk imkmrca! z}mam zmr e}jdh e²cz jhepaljcbh+



2+3+ Prlell~cz :

R²MJKLJCZ BM LKRMI_CJL ²HK

Cs}²ĳ ~mrmehz ac ²mjkljcz e²cz jhe}kmz pcrc jcaj}acr ac lkmirca $bmffiklbc h lkbmffiklbc,

bm }kc n}kjl²hk+

Prlell~cz o²czljcz

tk 5

tk&3

k & 3& j! k 5 ∟3

bt

t5 ak qtq& j

mct 5

3

cmct & j

zmk t 5 ∟ jhz t & j

jhz t 5 zmk t & j

3

jhz? t5 ck t & j

3

zmk? t5 ∟ jh t & j

3

t? & c?5

3

ccrjck

t

c

& j

3∗

c? ∟ t?5 crjzmk

tc

& j

zmkd t 5 jhzd t & j

jhzd t 5 zmkd t & j

Lkmircjl²hk emblckm jceolh bm ~crlcoamz h phr z}ozl}jl²hk $J^,

Lkmirca bmffiklbc i$o,i$c,

n $t,bt 5

oc

n $i$,,i$,b

Lkmirca lkbmffiklbc n $t,bt 5

n $i$,,i$,b

ca ffikca! dc| s}m bmzdcjmr ma jceolh

Lkmircjl²hk phr pcrmz $LPP,

Lkmirca bmffiklbc oc

n $t,i$t,bt 5 n $t,i$t,oc∟ oc

n $t,i$t,bt

Lkmirca lkbmffiklbc n $t,i$t,bt 5 n $t,i$t,∟

n $t,i$t,bt

}b~ 5 }~ ∟

~b}



2 Lkmircjl²hk

Lkmircjl²hk bm n}kjlhkmz rcjlhkcamz7 bmzjhephzljl²hk mk nrcjjlhkmz zlepamz

P $t,

S$t,bt ↟ P! S phalkhelhz+

Zl ma ircbh bm P ≭ ircbh bm S ⇟ bmomehz bl~lblr ahz phalkhelhz7P $t, 5 S$t,J $t, & _$t, ↟

P $t,

S$t,bt 5

J $t, &

_$t,

S$t,bt+

_$t,

S$t,bt jhk ircbh$_$t,, 1 ircbh$S$t,,7

l, Prlemrh! bmomehz jheprhocr s}m ac lkmirca kh zmc lkemblcc! mz bmjlr! zlmz bma lph7

lph ak ↟

?t & :

t? & :t & 4bt 5 ak qt? & :t & 4q & j+

lph crjck ↟

bt

t? & 45

3∗4

crjckt∗

4& j+

ll, Zl kh ah mz ↟ Dcrmehz bmzjhephzljl²hk mk nrcjjlhkmz zlepamz7

Ncjhr mk ma bmkhelkcbhr R²mrelkh mk ac bmzjhephzljl²hk mk nrcjjlhkmz zlepamz

t∟ oC

t∟ o

$t∟ o,fC3

t∟ o&

C?

$t∟ o,?& ¹ ¹ ¹ Cf

$t∟ o,f! f 5 3! ?! :! ¹ ¹ ¹

$t∟ c,? & o?Ct & O

$t∟ c,? & o?

$t∟c,?&o?

f C3t & O3

$t∟ c,? & o?&

C?t & O?$t∟ c,? & o?

? &¹ ¹ ¹& Cft & Of$t∟ c,? & o?

f ! f 5 3! ?! :! ¹ ¹ ¹

Pcrc jcbc ncjhr mk ma bmkhelkcbhr bmomehz ckˑcblr ma ²mrelkh jhrrmzphkblmkm bm accoac | jcaj}acr acz lkj²hiklcz $C!O!C3! O3! C?! O?! ¹ ¹ ¹ , li}cackbh ahz bmkhelkcbhrmz+Bmzp}²mz dc| s}m jcaj}acr ac lkmirca bm jcbc ²mrelkh+

C pcrlr bm cs}²ĳ! _ 5P

Sbmkhc }kc n}kjl²hk rcjlhkca mk z}z cri}emkhz! P! S zhk

phalkhelhz+



2+3+ Prlell~cz =

N}kjlhkmz lrrcjlhkcamz h lkmircamz jhk rc²ĳjmz

Dcrmehz }k jceolh bm ~crlcoamz pcrc malelkcr acz rc²ĳjmz+ _

ct & o

jt & b

p3(s3! ¹ ¹ ¹ !

ct & o

jt & b

pr(sr↟ e 5ct & o

jt & b! e 5 eje$s3! ¹ ¹ ¹ ! sr,+

eje↟ e²ĳkleh jhe²}k e²}alpah+

Lkmircamz bm n}kjlhkmz rlihkhe²mrljcz

zmk?k t!

jhz?k t ↟ n²hre}acz bm ²cki}ah bhoam7 jhz ?t 5 jhz? t ∟ zmk? t

zmk?k&3 t 5 zmk?k t zmk t 5 $3

∟jhz? t,k zmk t

jhz?k&3 t 5

jhz?k t jhz t 5

$3 ∟ zmk? t,k jhz t zmk et jhz kt ↟ n²hre}acz rlihkhe²mrljcz

_$zmk t! jhz t, ↟

_ lepcr mk zmk t ↟ 5 jhz t_ lepcr mk jhz t ↟ 5 zmk t_ pcr mk jhz t | zmk t ↟ 5 ck t_mzh bm prhoamecz ↟ 5 ck t(?!

zmk t 5?

3 & ?! jhz t 5

3∟ ?

3 & ?! bt 5

?

3 & ?b

Cai}khz jceolhz bm ~crlcoamz

3+

_$t!

t? & c?, ↟ t 5 c ck

?+

_$t!

t? ∟ c?, ↟ t 5

c

jhz

:+

_$t!

c? ∟ t?, ↟ t 5 c zmk



> Lkmircjl²hk

2+?+ Ma Rmhrmec N}kbcemkca bma J²caj}ah

Zmc n lkmircoam mk Vc!oU! N $t, 5 tc

n $,b mz }kc prlell~c bm n $t, bmffiklbc mk Vc! oU+

Rmhrmec 2+?+3n lkmircoam mk Vc! oU ⇟ N jhklk}c mk Vc! oU

Rmhrmec 2+?+? $Ma Rmhrmec N}kbcemkca bma J²caj}ah! RNJ,

Zmc n lkmircoam mk Vc! oU | N $t, 5

tc

n $,b! bmffiklbc ∀t ∄ Vc! oU+

Zl n mz jhklk}c mk j ∄ Vc! oU ⇟ N mz bmrl~coam mk j | N $j, 5 n $j,+

Zl n mz jhklk}c ∀t ∄ Vc! oU ⇟ N mz bmrl~coam ∀t ∄ Vc! oU | N $t, 5 n $t,+

Rmhrmec 2+?+: $_miac bm Ocrrhy, Zmck n | i jhklk}cz mk Vc! oU | i bmrl~coam mk $c! o,! bm nhrec s}m i$t, 5 n $t,! ∀t ∄ $c! o,! mkhkjmz

oc

n 5

oc

i 5 i$o, ∟ i$c,+

Rmhrmec 2+?+2 $RNJ imkmrcalwcbh, Zmc N $t, 5

tc

n ! jhk n lkmircoam

Zmc D $t, 5 N $i$t,, 5

i$t,c

n ! mkhkjmz zl i mz bmrl~coam! mkmehz s}m

D $t, 5 N $i$t,,i$t, 5 n $i$t,,i$t,+

Zmc D $t, 5

i$t,a$t,

n ! mkhkjmz zl i | a zhk bmrl~coamz! mkmehz s}m

D $t, 5 n $i$t,,i$t, ∟ n $a$t,,a$t,+



2+:+ Cpaljcjlhkmz bm ac Lkmirca 9

2+:+ Cpaljcjlhkmz bm ac Lkmirca

²C_MCZ

²Crmc mkrm ac ir²cffijc bm }kc n}kjl²hk | ma mgm t! mkrm c | o7

C 5

ocqn qbt

²Crmc mkrm acz ir²cffijcz bm bhz n}kjlhkmz n! i! mkrm c | o7

C 5

ocqn ∟ iqbt

²Crmc jhk mj}cjlhkmz pcrce²mrljcz7 ma ²crmc mkrm ac ir²cffijc bm t 5 t$,! | 5 |$,| ma mgm t mkrm 5 0 | 5 3 mz7

C 5

30

|$,t$,b

²Crmc mk jhhrbmkcbcz phacrmz7 ma ²crmc bm ac ir²cffijc bm r 5 r$δ, mkrm δ 5 γ |δ 5 ο mz

C 5

ο γ

3

?r?$δ,bδ

^HA²]EMKMZ

^ha}emk phr zmjjlhkmz pcrcamacz7 zl C$t, mz ma ²crmc bm acz zmjjlhkmz pcrcamaczc ah acrih bm hbc ac ahkil}b bm }k z²halbh! ma ~ha}emk bma z²halbh cz²ĳ bmffiklbh mkrmt 5 c | t 5 o mz

^ 5

oc

C$t,bt

Ma e²mhbh bm blzjhz7 ma ~ha}emk bm }k z²halbh bm rm~ha}jl²hk homklbh ca rhcrqn $t,q carmbmbhr bma mgm t mkrm t 5 c | t 5 o mz

^ 5

oc

π$n $t,,?bt

Ma e²mhbh bm jcpcz7 ma ~ha}emk bm }k z²halbh bm rm~ha}jl²hk homklbh ca rhcrn $t, ≭ 0 carmbmbhr bma mgm | mz

^ 5 ?π

oc

tn $t,bt



4 Lkmircjl²hk

AHKILR]BMZ

Ac ahkil}b bma crjh bm j}r~c n $t, mkrm t 5 c | t 5 o mz

A$n , 5

oc

3 & $n $t,,?bt

Zl ac j}r~c ~lmkm bcbc mk nhrec pcrce²mrljc! ac ahkil}b mz

A 5

30

$t$,,? & $|$,,?b

LKRMI_CA LEP_HPLC

Bmffikljl²hk 2+:+3 Ac zli}lmkm lkmirca

∛

cn $t, 5 a²ĳe

K ↟∛

K

cn $t,!

zm aacec }kc lkmirca leprhplc bm n + Zl ma a²ĳelm mz ffiklh bmjlehz s}m ac lkmirca

jhk~mrim mk hrh jczh! blrmehz s}m ac lkmirca bl~mrim+

Rmhrmec 2+:+? $Jrlmrlh lkmirca pcrc zmrlmz, Jhkzlbmrmehz n ≭ 0 }kc n}kjl² hk

ehk² hhkc bmjrmjlmkm bmffiklbc mk t

≭3+ Zmc ck 5 n $k,! mkhkjmz

∛k53

ck |

∛

3n $t, bt!

lmkmk ma elzeh jhephrcelmkh! h ceocz jhk~mrimk h ceocz bl~mrimk+

Documents

Apuntes de Matemáticas para la Economía I