How to get rich. One hour math. The Deal! Example. Come on! Solution part 1: Constant income, no loss. by Stefan Trapp
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1 O hour h by Sf Trpp How o g rich Th Dl! offr you: liflog, vry dy Kr for o-i py ow of oly 5 Kr. d irs r of % bu oly o h oy you hv i.. h oy gv you ius h oy you pid bc for h irs No d o py bc yhig ls! s h dl? Expl Dy : Kr - r *% + ico Kr = Kr Dy : Kr - %* Kr + Kr =.6 Kr Dy :.6 Kr + Kr = 5.6 Kr - % of.6 Kr = 4.88 Kr d so o. si-ifiily s log s liv Co o! M h dl d g rich Soluio pr : Cos ico, o loss You r youg d g lif-log oy! Thr is o uch o los bu so uch o wi Who ccps r hr Expl Δ / Δ gi: you g cos ico of = Kr pr dy How uch oy do you hv - fr dys? - fr dys? - fr dys? how do you clcul his?
2 oy oy Kr Wh is ss blc? Expl Δ / Δ gi: you g cos ico of = Kr pr dy i h Δ / Δ = = Kr/dy Δ / Δ is h chg of oy Δ i your poc pr i priod Δ Th oy you hv fr i Δ is h iiil oy plus Δ D = + Δ Th oy blc = + Δ / Δ * is h iiil oy Kr Δ / Δ is h slop of h li hr: Kr/d = cos i d Expl: pu wih dcy r You g your oy = Kr / d bu you hv o py irs = % pr dy for h ou of oy you lrdy hv + pu: Kr/d How rich do you g, filly? - loss: % of Kr = Kr Kr i your poc i blcs sorry pu wih dcy r xcl soluio 5 5 = i d No so rich! Your ico soo blcs h xs! You d up wih sowh li Kr i your poc. Th soluio ws iriv d usd Δ d Δ, d EXCEL pu wih dcy r diffril quio + = + D = + - How o solv h probl sir wih h! f you Δ vry vry sll -->, h D li D D d d fro h diffrc o h diffril d d pu wih dcy r how o solv? d d This is lir is s ordr ordiry oly o dx/d o-hoogous wih ipu r diffril quio d How o solv his? igr --> lyicl soluio b uricl soluio
3 pu wih dcy r - soluio d d Grl Soluio This yp of quio hs lwys h soluio This is our firs ipor diffril quio Dfiiio lyicl soluio Th igrio of h diffril quio d d givs h lyicl soluio : Wih h rsulig quio, w c clcul idily h xc vlu of h vribl ll is d for ll iiil codiios. Th lriv is h spwis, pproxi uricl soluio. pu wih dcy r sdy-s Sdy-s s o chg wih i d d follows our xpl: = Kr d -, =. d - --> = Kr This will b h fil ou of oy you g fro pu wih dcy r Sdy-s scod soluio hod: --> follows Lir DE r lwys sdy-s for --> Co o! No, do rus hos offrs M h dl d g rich Thr is o such hig s fr luch i lif! You r youg d g lif-log oy! bu h sdy-s cos fr dys Thr is o uch o los oly 4 Kror. Thy r i ow bu so uch o wi oly for, hh You ow oy if you ccpd h dl Propris of lir ordiry diffril quios lir ODEs
4 ss Lir ODEs co lwys o sdy-s for / for : =, =., =, / = : =, =., =, / = : =, =., =, / = 4: = 5, =., =, / = / = 5 5 Ti 4 Dfiiio Hlf-i Th i uil hlf of h ou is rovd A hlf-i : =.5 Equio: = - Clculio: / = - = ½ Rsul: -l = ½ = l/ l Syoys: hlf-lif; / ; 5% ; Dfiiio Hlf-i = =, =., / =.466 l A i = hlf-i /, = ½ Ti o sdy-s hory, sdy-s is oly rchd wh = prxis, wih 6% rror, sdy-s is rchd fr bou 4 hlf-livs * : 5% * : 75% * : 87.5% 4 * : 9.75% 5 * : % 6 * : % cr l l l 4l 4 4l 4l % 94% sdy-s = 4 Pricipl of Suprposiio Cocrios c b ddd Curv : =., =.466; 4 =.86 Curv : =.5, =.86; 4 = 55.5 givs his 8 6 his 4 d his Moy c b ddd, ss c b ddd, sugr... Eq Eq Eq +Eq Eq Diffril quio c b ddd 4
![chg of ss = ss i - ss ou = D w ss = old ss + chg of ss = + D = ss [g or g] = ipu [g/d] = r [/d] = i [d] / = [g/d /d - ] = [g] Uis * = /d * d = disiolss --> uis r cosis.](/docs-images/81/82973586/images/5-1.jpg "Th quios ri h s = Fro ss o cocrio Fro ss o cocrio Mss : Ui g or g or g... C = /M or C = /V Cocrio C: Ui g/ or g/l or g/g.")
5 Pricipl of Suprposiio Prs of DE c b solvd idividully, d b ddd Eq. d d Eq. d, for _ d d Eq. su d Ay qusios? 8 6 q. =+ q. 4 q Eq Eq Eq +Eq Eq Fro oy o ss Now rplc oy by ss of chicl d i's ss blc! chg of ss = ss i - ss ou = D w ss = old ss + chg of ss = + D = ss [g or g] = ipu [g/d] = r [/d] = i [d] / = [g/d /d - ] = [g] Uis * = /d * d = disiolss --> uis r cosis. Th quios ri h s = Fro ss o cocrio Fro ss o cocrio Mss : Ui g or g or g... C = /M or C = /V Cocrio C: Ui g/ or g/l or g/g... C = /V liquids whr V is volu or L C = /M solids whr M is ss of solid g follows h dc x V/d = d/d = - For cos V, w c wri V x dc/d = - d dc/d = /V - /V = /V - C wih h ow coo soluio C C V 5
6 % Cocrios i chisry pp ppb pp ppq Now y qusios? g/g g/g g/g pg/g Th big h solvrs Soluio : Supid, bu good figrs h copur Soluio : by brilli hici Aro Ri Sp-wis uricl soluio Df: fro h diffrc o h diffril D d li D D d Th lso d D li d D D d hus d D d D wih dl diffrc w c clcul D D d g h w vlu for h x i sp D D d so o, sp for sp... Eulr plio i xcl Advgs d disdvgs of uricl hods Nuricl soluio Esy o copur My b wrog For vribl cofficis Fr ipu fucio C i Difficul o chc clculio ds copur Alyicl soluio Esy by hd Alwys xc For cos cofficis oly Oly fw ipu fucios Exiss oly for spcil probls Trspr ds bri Good for prcicl wor Good for ducio 6
7 Coupld diffril quios by Sf Trpp & Aro Ri Fro o o y diffril quios Uil ow w hd lwys o vribl ss or cocrio C for which w d d solvd o diffril quio. Syss of diffril quios = DE s h hr is or h o DE. Expl : Cscd of s or rcors Cscd - coupld diffril quio d d d d Mss blc: Th chg of ss i is wh flows ou of ius wh flows ou of. d d d d Grl soluio for = lir DGL d ipu is lwys!: iiil ou This is coupld lir diffril quio sys d d Two coupld diffril quio d d cscd cscd 4 5 pu Cscd - digol rics Expl Wr - Sdi loss Wr xchg o o wy bc Sdi loss 7
8 8 Th Mrix sys of DE Trdiiol quio wri s rix d d d d d d d d hicis us: Cscd = Alyicl soluio for ipu oly Cscd = d ipu C Ti d C C C Expl: =, =.5, =. d - Alyicl soluio hod = Grl soluio for cscd sys wih ipu io : Loos coplicd, d prssiv, bu i fc i is rhr sy o ipl., i i Cscd = Suprposiio pricipl: Alyicl soluio for ipu,, iiil ss, Alyicl soluio hod = Grl soluio for cscd sys wih puls ipu io ll coprs: i i, Suprposiio of soluios wih ipu plus plus c.
9 C C Expl : Cscd of s wih hol Cscd wih hol rix forulio For 4 rix ls, i.. 4 coprs, h rix is giv by: d d d d As bfor, bu wih ddiiol loss by hols i h. Mss blc: Th chg of ss i is wh flows ou of h hol i d wh flows io. Th chg of ss i is wh flows ou of h hol i d wh flows io. Df.: = + ol loss r = + whr dc d 4 C b 4,, 4 is h ol loss r fro copr,,4;, 4 is h rsfr r fro cop. o o c. wih d b is h ipu vcor = / V. Such digol rics c lso b solvd lyiclly, dpdig o h iiil codiios d h ipu fucio Trpp d Mhis 998. Cscd wih hol puls ipu Cscd copriso wih / ou hol Alyicl soluios for C C C C C C C C C C Alyicl soluios for C C C C C Sll bu ipor diffrc. T cr! ll loss fro gos io oly loss gos io Cscd copriso wih / ou hol PCE --. d - -> TCE --. d - -> DCE --. d - -> VC --. d - -> Eh Cscd wih hol puls ipu, grl Grl lyicl soluios for C.E+.E+ 8.E+ 6.E+ 4.E+.E+.E+ 5 5 Rl i d.e+.e+ 8.E+ 6.E+ 4.E+.E+.E+ 4 6 C C i, i C i, > PCE TCE DCE VC Rl i d Lf sid: oly bolis PCE TCE DCE VC Righ sid: bolis + volilisio. d - All cscd chis wih lgh c b solvd lyiclly bu wih highr ubr i gs roublso! Wih ddiiol si r. d - for ll copouds PCE d bolis vish uch fsr 9
10 C Cscd wih hol, cos ipu For 4 rix ls, h rix is dc d 4 C b 4 whr b is h ipu vcor b, b, b d b 4, so h dc C b d dc C C b c. d Cscd wih hol d cos ipu Sdy-s soluio S dc/d lf hd o zro. Th C c. b M C C M C C M Coc. = pu / loss Cscd wih cos ipu Alyicl soluio for ll b C C A B C C D E F C C C b A b b B D A E C A B B b F Expl PCE dgrdio uoril Cos ipu for so i, h oly loss.e+.5e+.e+.5e+.e+ 5.E+ cos.e Rl i d puls ipu PCE TCE DCE PCE TCE DCE No-digol rics Expl Wr - Sdi Th full x rix Cosidr h wo coupld diffril quios loss pu Wr xchg o xchg Sdi loss This diffrs fro ll our rlir xpls bcus hr is rsfr fro copr o d bc fro o. dc d dc d C b C C b C or, i rix for dc C b d wih = loss + rsfr d = loss + rsfr This diffrs fro ll our rlir xpls bcus hr is rsfr fro copr o d bc fro o.
11 C Fro WN Nzroff d L Alvrz-Coh: Evirol girig scic Joh Wily & Sos Nw Yor,, pg Br loo up hr, oo! Ts x full rix Eigh rs r dd T / 4 T / 4 T C C T T T T 4 4 C T C TT T5 4 T6 C C T T T6 T7 4 C T C TT6 T C C Ti C C C T T6 T7 T8 T C T T T4 T5 T Oc you hv dfid h 8 rs i s vry usful d sipl soluio hod, bliv i los vr fils! How did fid ll hs soluios? Rd boos or our scrip. Th sdrd lyicl soluios r ll ow d wri dow. gr. Firs for, h his io quio for c. Usig highr h s scrip, pgs o 7: Th hoogous lir DE sys hs soluio if d A - l E = s scrip whr d is dri; E is ui rix l is Eigvlu roos of h chrcrisic polyo 4 s Aro Ri! y bs ric Whr r which soluios usd Sdy-s: - Roo odl, Frui r odl 7, poo odl 7, TGL odl 995 Sdy-s d lir DE: - Sdrd odl Coupld DE "Cscd wih hol d cos ipu": - Cscd odl - Bucs odl sorry for h y h... Qusios?
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