Chapter 7 Rules of Differentiation & Taylor Series

Transcription

1 RS - Ch 7 - Ruls o Dirtiatio Chaptr 7 Ruls o Dirtiatio & Taylor Sris Isaa Nwto a Gottri Libiz 7. Rviw: Drivativ a Drivativ Ruls Rviw: Diitio o rivativ. y ' lim lim Applyig this iitio, w rviw th 9 ruls o irtiatio. First, th ostat rul: y Th rivativ o a ostat utio is zro or all valus o. y y k k y ' lim I k th k lim k k lim

2 RS - Ch 7 - Ruls o Dirtiatio 7. A Rviw o th 9 Ruls o Dirtiatio 5a 5b k g g g g g Costat a prout Costat, Costat Powr Sum - ir Prout utio prout, utio rul ruls & powr ruls 7. A Rviw o th 9 Ruls o Dirtiatio 6 7-8:Lt : Lt 9 g z g g,..., y, y g g g y y y whr y is a stritly mootoi utio o y g z y y.. g z y y, a Uivariat Chai rul Multivariat Chai rul y Quotit rul Ivrs- utio rul

3 RS - Ch 7 - Ruls o Dirtiatio Powr-Futio Rul lim... lim... lim lim lim ' Eampl: Lt Total Rvu R b: R = 5 Q Q => RQ = MR =5 Q. As Q irass R irass as log as Q> Epotial-Futio Rul h h h h κ h κ h h ' lim Thus, lim.th, Lt uiqu positiv umbr or whih lim : Diitio o lim lim lim t t t t t y Eampl Growth : Epotial

4 RS - Ch 7 - Ruls o Dirtiatio 7.. Epotial-Futio Rul: Jok A mathmatiia wt isa a bliv that h was th irtiatio oprator. His ris ha him pla i a mtal hospital util h got bttr. All ay h woul go arou rightig th othr patits by starig at thm a sayig "I irtiat you!" O ay h mt a w patit; a tru to orm h star at him a sai "I irtiat you!", but or o, his vitim's prssio i't hag. Surpris, th mathmatiia ollt all his rgy, star irly at th w patit a sai louly "I irtiat you!", but still th othr ma ha o ratio. Fially, i rustratio, th mathmatiia sram out "I DIFFERENTIATE YOU!" Th w patit almly look up a sai, "You a irtiat m all you lik: I'm." 7.. Sum or ir rul Th rivativ g g o a sum or ir o sum or ir o th rivativs o two utios is th th two utios Eampl : C Q C Q Q C Q Q Q Q Q Q 8Q Q 75 Q Q Q 75 8

5 RS - Ch 7 - Ruls o Dirtiatio 7.. Prout rul g g g Th rivativ o th prout o two utios is qual to th so utio tims th rivativ o th irst plus th irst utio tims th rivativ o th so. Eampl: Margial Rvu R PQ R Q Chk : Q Q R Q P 5 -Q P P Q 5 Q 5 Q Q Q R 5 -QQ 5 Q R Q 5 Q Quotit rul 6 g g g g Eampl : TC CQ AC CQQ C Q Q i Q Q Q C Q C Q Q C Q, th Q Total ost Avrag ost C Q Q Q C Q Q AC MC Avrag Cost Margial Cost MC AC 5

6 RS - Ch 7 - Ruls o Dirtiatio 7.. Chai rul This is a as o two or mor irtiabl utios, i whih ah has a istit ipt variabl, whr z y, i.., z is a utio o variabl y a y 7 g, i.., y is a utio o variabl z z y y y yg Q y L Eampl: R Q rvu& R R Q L Q L g MR MPP L MRP L y g z Q gl g.that is, output 7.. Chai rul: Appliatio - Log rul Chai Rul : z z y y yg y y y g Cosir h = l = Th, h =. Lt s apply Chai rul to h h ' l l l l l 6

7 RS - Ch 7 - Ruls o Dirtiatio 7.. Chai rul a its rlatio to total irtial Fi z, whr z y a y g, z Algorithm : Substitut th total irtial o y ito that o z a ivi through by z y y y y y assumig z z z y. y z y y y 7.. Ivrs utio rul Lt y= b a irtiabl stritly mootoi utio. y y y Not: A mootoi utio is o i whih a giv valu o yils a uiqu valu o y, a giv a valu o y will yil a uiqu valu o a o-to-o mappig. Ths typs o utios hav a i ivrs. Eampl : Q P Q s Q s b P b P b P P PQ -b - s Q b b b Q s [ Q s whr P] b 7

8 RS - Ch 7 - Ruls o Dirtiatio 7.. Ivrs-utio rul This proprty o o-to-o mappig is uiqu to th lass o utios kow as mootoi utios: Rall th iitio o a utio utio: o y or ah mootoi utio: o or ah y ivrs utio i > > mootoially irasig Q s = b + b P supply utio whr b > P = -b b + b Q s ivrs supply utio i > < mootoially rasig Q = a -a P ma utio whr a > P = a a -a Q ivrs ma utio 5 7. So a Highr Drivativs Drivativ o a rivativ Giv y = Th irst rivativ ' or y is itsl a utio o, it shoul b irtiabl with rspt to, provi that it is otiuous a smooth. Th rsult o this irtiatio is kow as th so rivativ o th utio a is ot as '' or y. Th so rivativ a b irtiat with rspt to agai to prou a thir rivativ: ''' a so o to or y This pross a b otiu to prou a -th rivativ. 6 8

9 RS - Ch 7 - Ruls o Dirtiatio 7. Eampl: st. a r rivativs Graphially: Q Q Q R Q Q Q Q R Q Q primitiv utio st rivativ rivativ r rivativ MR Q Q Q MR 7 7. Eampl: Highr Orr Drivativs 5 6 y ' 6 " primitiv utio st rivativ rivativ r rivativ th rivativ 5th rivativ 8 9

10 RS - Ch 7 - Ruls o Dirtiatio 7. Itrprtatio o th so rivativ ' masurs th rat o hag o a utio.g., whthr th slop is irasig or rasig '' masurs th rat o hag i th rat o hag o a utio.g., whthr th slop is irasig or rasig at a irasig or rasig rat how th urv ts to b itsl Utility utios ar irasig i osumptio '>. But thy ir by th rat o hag i '>; that is, thy ir o ''. '' >, irasig '> '' =, ostat '> '' <, rasig '> usual assumptio 9 7. Strit oavity a ovity Stritly oav: i w pik ay pair o poits M a N o its urv a joit thm by a straight li, th li sgmt MN must li tirly blow th urv, pt at poits MN. A stritly oav urv a vr otai a liar sgmt aywhr i it os it s just oav, ot stritly oav. Tst: i " is gativ or all, th stritly oav. Stritly ovity: i w pik ay pair o poits M a N o its urv a joit thm by a straight li, th li sgmt MN must li tirly abov th urv, pt at poits MN. A stritly ov urv a vr otai a liar sgmt aywhr i it os it s just ov, ot stritly ov. Tst: i " is positiv or all, th stritly ov.

11 RS - Ch 7 - Ruls o Dirtiatio Figur 7.6 Coav a Cov Futios 7. Coavity a Covity: & I " < or all, th stritly oav. a global maima I " > or all, th stritly ov. a global miima Coav utios hav valuabl proprtis: ritial poits ar global maima, & th wight sum o oav utios is also oav. A popular hoi to srib a avrag utility a proutio utios. Eampl: AP = Arrow-Pratt risk avrsio masur = -U wu w Lt Uw = β lw β > U w = βw > U w = -β w - < AP = w As w walth irass, risk avrsio rass.

12 RS - Ch 7 - Ruls o Dirtiatio Figur 7.5 Logarithmi Utility Futio Figur 7.7 Utility Futios or Risk- Avrs a Risk-Lovig Iiviuals

13 RS - Ch 7 - Ruls o Dirtiatio 7.5 Sris Diitio: Sris, Partial Sums a Covrg Lt {a } b a iiit squ.. Th ormal prssio Σ a is all a iiit sris.. For N =,,,... th prssio S = Σ a is all th N-th partial sum o th sris.. I lim S ists a is iit, th sris is sai to ovrg.. I lim S os ot ist or is iiit, th sris is sai to ivrg. Eampl: Σ = a iiit sris Th r, a th partial sums ar, rsptivly:.875, a.975. Th -th partial sum or this sris is i as S = Divi S by a subtrat it rom th origial o, w gt: S - S = - + S = Th, lim S = th iiit sris ovrgs to 7.5 Sris: Covrg A sris may otai positiv a gativ trms, may o thm may al out wh a togthr. H, thr ar irt mos o ovrg: o mo or sris with positiv trms, a aothr mo or sris whos trms may b gativ a positiv. Diitio: Absolut a Coitioal Covrg A sris Σ a ovrgs absolutly i th sum o th absolut valus Σ a ovrgs. A sris ovrgs oitioally, i it ovrgs, but ot absolutly. Eampl: Σ - = => o absolut ovrg Coitioal ovrg? Cosir th squ o partial sums: S = = - i is o, a S = = i is v. 6 Th, S = - i is o a i is v. Th sris is ivrgt.

14 RS - Ch 7 - Ruls o Dirtiatio 7.5 Sris: Rarragmt Coitioally ovrgt squs ar rathr iiult to work with. Svral opratios o ot work or suh sris. For ampl, th ommutativ law. Si a + b = b + a or ay two ral umbrs a a b, positiv or gativ, o woul pt also that hagig th orr o summatio i a sris shoul hav littl t o th outom Thorm: Covrg a Rarragmt A sris Σ a b a absolutly ovrgt sris. Th ay rarragmt o trms i that sris rsults i a w sris that is also absolutly ovrgt to th sam limit. Lt Σ a b a oitioally ovrgt sris. Th, or ay ral umbr thr is a rarragmt o th sris suh that th w rsultig sris will ovrg to Sris: Absolut Covrgt Sris Absolutly ovrgt sris bhav just as pt. Thorm: Algbra o Absolut Covrgt Sris Lt Σ a a Σ b b two absolutly ovrgt sris. Th:. Th sum o th two sris is agai absolutly ovrgt. Its limit is th sum o th limit o th two sris.. Th ir o th two sris is agai absolutly ovrgt. Its limit is th ir o th limit o th two sris.. Th prout o th two sris is agai absolutly ovrgt. Its limit is th prout o th limit o th two sris Cauhy Prout. 8

15 RS - Ch 7 - Ruls o Dirtiatio 7.5 Sris: Covrg Tsts Thr ar may tsts or ovrg or ivrg o sris. Hr ar th most popular i oomis. Divrg Tst I th sris Σ a ovrgs, th {a } ovrgs to. Equivaltly: I {a } os ot ovrg to, th th sris Σ a a ot ovrg. Limit Compariso Tst Suppos Σ a a Σ b ar two iiit sris. Suppos also that r = lim a b ists a < r <. Th Σ a ovrgs absolutly i Σ b ovrgs absolutly Sris: Covrg Tsts p Sris Tst Th sris Σ p is all a p Sris. i p > th p-sris ovrgs i p th p-sris ivrgs. Altratig Sris Tst A sris o th orm Σ - b, with b is all altratig sris. I {b } is rasig a ovrgs to, th th sum ovrgs. Gomtri Sris Tst Lt a R. Th sris Σ a is all gomtri sris. Th, i a < th gomtri sris ovrgs i a th gomtri sris ivrgs. 5

16 RS - Ch 7 - Ruls o Dirtiatio 7.5 Sris: Powr Sris Diitio: Powr Sris A utio sris o th orm Σ a - = a + a - + a is all a powr sris tr at. That is, a powr sris is a iiit sris o utios whr ah trm osists o a oiit a a a powr -. Eampls: - Σ = + = Σ = - = Σ = - = Popular ampl i Fia DDM isout ivi mol: 7.5 Sris: Powr Sris Proprtis: - Th powr sris ovrgs at its tr, i.. or = - Thr ists a r suh that th sris ovrgs absolutly a uiormly or all - p, whr p<r, a ivrgs or all - > r. r is all th raius o ovrg or th powr sris a is giv by: r = lim sup a a + Not: It is possibl or r to b zro --i.., th powr sris ovrgs oly or = -- or to b -i.., th sris ovrgs or all. Eampl: Σ = - r = lim sup a a + = lim sup + + = lim sup +* = Sris ovrgs absolutly a uiormly o ay subitrval o - <. 6

17 RS - Ch 7 - Ruls o Dirtiatio 7.5 Sris: Powr Sris Polyomials ar rlativly simpl utios: thy a b a, subtrat, a multipli but ot ivi, a, agai, w gt a polyomial. Dirtiatio a itgratio ar partiularly simpl a yil agai polyomials. W kow a lot about polyomials.g. thy a hav at most zros a w l prtty omortabl with thm. Powr sris shar may o ths proprtis. Si w a a, subtrat, a multiply absolutly ovrgt sris, w a a, subtrat, a multiply thik Cauhy prout powr sris, as log as thy hav ovrlappig rgios o ovrg. Dirtiatig a itgratig works as pt. Importat rsult: Powr sris ar iiitly ot lim sup irtiabl. 7.6 Taylor Sris Th Taylor sris is a rprstatio o a iiitly irtiabl utio as a iiit sum o trms alulat rom th valus o its rivativs at a sigl poit,. It may b rgar as th limit o th Taylor polyomials. Brook Taylor 685 7, Egla Diitio: Taylor Sris Suppos is a iiitly ot irtiabl utio o a st D a D. Th, th sris T, = Σ [!] - is all th ormal Taylor sris o tr at, or arou,. I =, th sris is also all MaLauri Sris. 7

18 RS - Ch 7 - Ruls o Dirtiatio 7.6 Taylor Sris: Rmarks - A Taylor sris is assoiat with a giv utio. A powr sris otais i priipl arbitrary oiits a. Thror, vry Taylor sris is a powr sris but ot vry powr sris is a Taylor sris. - T, ovrgs trivially or =, but it may or may ot ovrg aywhr ls. I othr wors, th r o T, is ot ssarily gratr tha zro -Ev i T, ovrgs, it may or may ot ovrg to Eampl: A Taylor Sris that os ot ovrg to its utio = p- i = i = Th utio is iiitly ot irtiabl, with =. T g, arou = has raius o ovrg iiity. T g, arou = os ot ovrg to th origial utio T g, = or all Malauri Sris: Powr Sris Drivatio a a a a a a... a a a a 6a... a 6a... a a 6a a a a! a Evaluatig ah utio at, simpliyi g & solvig or th oii t!! a! a...! a! a! a primitiv utio rivativ a a! a! a! Substituti g th valu o th oii ts ito th primitiv utio st r th rivativ rivativ rivativ!!! 6 8

19 RS - Ch 7 - Ruls o Dirtiatio 7.6 Taylor Sris: Taylor s Thorm Suppos C + [a, b] -i.., is +-tims otiuously irtiabl o [a, b]. Th, or [a,b] w hav:! whrr!! p p p R I partiular, th T, or a iiitly ot irtiabl utio ovrgs to i th rmair R + ovrgs to as.!! W a show that a utio rally has a Taylor sris by hkig to that th rmair gos to zro. Lagrag ou a asir prssio: p R! 7 or som p btw a. 7.6 Taylor Sris: Taylor s Thorm Impliatios: - A utio that is +-tims otiuously irtiabl a b approimat by a polyomial o gr. -I is a utio that is +-tims otiuously irtiabl a + = or all th is ssarily a polyomial o gr. - I a utio has a Taylor sris tr at th th sris ovrgs i th largst itrval -r, +r whr is irtiabl. I prati, a utio is approimat by its Taylor sris usig a small, say =:!!! p with R! Th rror & th approimatio ps o th urvatur o. 8 9

20 RS - Ch 7 - Ruls o Dirtiatio 7.6 Taylor Sris: Taylor s Thorm - Eampl Taylor sris pasio o a quarati polyomial arou =. =5+ + = = 8 = + = = = = = = = = Taylor s sris ormula:!! For,! & First-orr Taylor sris: = R = + + R with R = [!]- =- p R!! Taylor Sris: Taylor s Thorm - Eampl Lt s hk th approimatio rror R =- : = R = = 8 = 8 =.. = 8.. = 8.. =.. = 8.8. = 8.8. For : R 8! 8! 5 R prt it R R R Not: Polyomial a b approimat with grat auray.

21 RS - Ch 7 - Ruls o Dirtiatio 7.6 Malauri Sris o Lt s o a Taylor sris arou =: primitiv utio st rivativ ' rivativ '' r rivativ ''' th rivativ Substitutig th valu o th oiits ito th primitiv utio!!!!! 7.6 Malauri Sris o os Lt s o a Taylor sris arou =: os primitivutio si os si os st r th rivativ rivativ rivativ rivativ os ' si '' os ''' si os Substitutig th valuo th oiits ito th primitivutio os!!! Now, lt s hk i th rmair R + gos to as : p os p R!!! a th last trm is a ovrgig sris to, as.

22 RS - Ch 7 - Ruls o Dirtiatio 7.6 Malauri Sris o si & Eulr s ormula Similarly, w a a Taylor sris or si: si! 5 7 5! 7!! Now, lt s go bak to th Taylor sris o. Lt s look at i : i i! i! i! os isi! 5 i i i i i!!! 5! 5! i i 5! This last rsult is all Eulr s ormula. It will r-appar wh solvig irtial quatios with ompl roots. 7.6 Malauri Sris o log + log! primitivutio st r th rivativ rivativ rivativ rivativ Evaluatigah utio at, simpliyig & solvig or th oiit log &!... Substitutig th valu o th oiits ito th primitivutio!!!! i i! i i

23 RS - Ch 7 - Ruls o Dirtiatio 7.6 Malauri Sris o log + log A st orr sris pasio: log+ = + O. Notatio: O : R is bou by A as or som A <. Blow, w plot log+ & its liar approimatio. Wh is small say, <., typial o aual itrst rats or mothly stok rturs, th liar approimatio works vry wll Taylor sris: Approimatios Taylor sris work vry wll or polyomials; th potial utio a th si a osi utios. Thy ar all ampls o tir utios i.., quals its Taylor sris vrywhr. Taylor sris o ot always work wll. For ampl, or th logarithm utio, th Taylor sris o ot ovrg i is ar rom. Log approimatio arou : 6

24 RS - Ch 7 - Ruls o Dirtiatio 7.6 Taylor Sris: Nwto Raphso Mtho Th Nwto Raphso NR mtho is a prour to itrativly i roots o utios, or a solutio to a systm o quatios, whr =. Vry popular i umrial optimizatio, whr =. To i th roots o a utio. W start with: W riv a itrativ pross to i th roots o a utio. ' ' ' ' y y k k k k k k k 7 k Xk+ Xk X B C A 7.6 Taylor Sris: NR Mtho 8 W a us th NR mtho to miimiz a utio. Rall that '* = at a miimum or maimum, thus statioary poits a b ou by applyig NR mtho to th rivativ. Th itratio boms: W '' k ; othrwis th itratios ar ui. Usually, w a a stp-siz, λ k, i th upatig stp o : k+ = k λ k ' k '' k. '' ' k k k k

25 RS - Ch 7 - Ruls o Dirtiatio 7.6 Taylor Sris: NR Quarati Approimatio Wh us or miimizatio, th NR mtho approimats by its quarati approimatio ar k. Epa loally usig a -orr Taylor sris: +δ = + ' δ + ½ '' δ + oδ +δ= +' δ+½ '' δ Fithδ whih miimizs this loal quarati approimatio: δ = ' '' Upat: k+ = k - ' ''. 7.6 Taylor Sris: Th Dlta Mtho Th lta mtho is us to obtai th asymptoti istributio o a o-liar utio o a raom variabl usually, a stimator. It uss a st-orr Taylor sris pasio a Slutsky s thorm. Lt b a RV, with plim =θ a Var =σ <. W kow th asymptoti istributio o : ½ - θσ N, But w wat to kow th asymptoti istributio o g. W assum g is otiuously irtiabl, ipt o. Stps: Taylor sris approimatio arou θ : g = gθ + gθ - θ + highr orr trms W will assum th highr orr trms vaish as grows. 5 5

26 RS - Ch 7 - Ruls o Dirtiatio 7.6 Taylor Sris: Th Dlta Mtho Us Slutsky thorm: plim g = gθ plim g = g θ Th, as grows, g gθ + gθ - θ ½ [g - gθ] gθ [ ½ - θ]. ½ [g - gθ]σ gθ [ ½ - θσ]. I g. os ot bhav baly, th asymptoti istributio o g - gθ is giv by that o [ ½ - θσ], whih is a staar ormal. Th, ½ [g - gθ] N, [gθ] σ. Atr som work ivrsio, w obtai: g Ngθ, [gθ] σ Taylor Sris: Th Dlta Mtho Eampl Lt a Nθ, σ Q: g =δ? δ is a ostat First, alulat th irst two momts o g : g = δ plim g =δθ g = -δ plim g =-δθ Rall lta mtho ormula: g Th, g a Nδθ, δ θ σ a Ngθ, [gθ] σ. 5 6

27 RS - Ch 7 - Ruls o Dirtiatio Gomtri sris !!!. arou Malauri sris th o trms irst iv th Fi. Giv a a a a Gomtri sris: Eah trm is obtai rom th prig o by multiplyig it by, ovrgt i <. 5 9 whr a a a a y. a a y 7.7 Gomtri sris: Approimatig -a - It is possibl to auratly approimat som ratios, with - trm i th omiator, with a gomtri sris. Rall: For =. a=. -a=. & =. a=.9 -a= & =.95

28 RS - Ch 7 - Ruls o Dirtiatio 7.7 Gomtri sris: Approimatig A - AA Taylor sris approimat io - I i I A A I A A... A by aalogy A A A A I... salar algbra Appliatio to th Loti Mol =A+ => I-A =.5 A..5.5 I A I A with 6 I A Appliatio: Gomtri sris & PV Mols A stok pri P is qual to th isout som o all uturs ivis. Assum ivis ar ostat a th isout rat is r. Th: whr. Eampl: = USD ; r = 8% P = USD.8 = USD.5 = 56 8

29 RS - Ch 7 - Ruls o Dirtiatio 7.7 Appliatio: Gomtri sris & PV Mols Now, w assum ivis grow at a ostat g a th isout rat is r. Th:, Eampl: = USD ; r = 8% & g = % P = USD *..8-. = USD 7. whr Not: NPV o ivi growth = USD 7 USD.5 = USD.5 57 Q: What is th irst rivativ o a ow? A: Prim Rib! 9