1. Mathematical tools which make your life much simpler 1.1. Useful approximation formula using a natural logarithm

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1 . Mhmicl ools which mk you lif much simpl.. Usful ppoimio fomul usig ul logihm I his chp, I ps svl mhmicl ools, which qui usful i dlig wih im-sis d. A im-sis is squc of vibls smpd by im. As mpl of ul l GD, vibl of l GD is smpd by 200, 20, 202, d so o. Fo quly l GD, i is smpd by sy, Juy-Mch, Apil-Ju, July-Spmb, d Ocob-Dcmb i 206. A gl pssio of im-sis vibl is, which is smpd by im. Fis of ll, I would lik o pick up o usful ppoimio fomul: If is clos ough o zo, h l( + ). A oio l is ul logihm, o logihm whos bs is Npi s cos. Npi s cos is iiol umb, d is of dod by ; = A ul logihm is h ivs fucio of poil fucio y = ; hus, i is dfid s = l y. L us us sciific clculo o compu l( ) ( ) l l ( ) l l ( ) l l ( ) l l l l fo svl vlus of. As h bov compuio dmoss, if h bsolu vlu of is lss h 0., h h ppoimio woks qui wll. Th umb you s i mcocoomic sisics, icludig h ul gowh of l GD d h ul of is

2 2 s, is of o-digi pcg. As you s l, h is why h ppoimio fomul l( + ) is qui usful. Bu, you my b sill doubful of how h ppoimio woks wihou sig is igoous poof. Fo his pupos, w hv o c bck o how Npi s cos is dfid. H is dfiiio of Npi s cos. lim + = Accodig o h bov dfiiio, o piod 00% u is dividd io sub-piods ( ), d compu compoud goss u fo o piod wh is lg ough. Th is Npi s cos! L us follow h sm pocdu fo cs wh o piod u is 00% isd of 00%; h is, lim +. Dfiig m s, w hv h followig gm: m m lim + = lim + = lim + = m m m m Wh sub-piod u is wih lg, cospods o compoud goss u fo o piod. O h oh hd, simpl goss u fo o piod is jus qul o + = +. W hv ow do ll h ppio fo povig l + if is clos o zo. W fis ppoim Th, i lds o:! + = 2! = = + = 2 oud zo by Tylo sis:! = + +. = 2 Wh is ppochig zo, covgs o o. Cosquly, + if is clos ough o zo. Th, kig ul logihms fo boh sids, w hv h ppoimio fomul: l = l ( + )

3 3 l GD Figu -: D o ul logihmic vicl is l 500 l 400 Slop = (l 500 l 400) /( ) y... Empl : Gowh ppoimio Now h poof fo h ppoimio fomul is compld, i is im o ploi i. L us bgi wih simpl pplicio. A gowh of X o X + c b ppoimd by l X+ l X if is gowh is clos ough o zo. A poof fo his ppoimio is qui simpl. X X X X X l X+ l X = l = l + X X X Th bov ppoimio hs ic gomic ipio. Suppos h l GD gw fom 400 uis i 200 o 500 uis i 205. Th ul gowh of l X X GD is compud by = ; h is, i gw by 4.56% p y. L us 400 k ul logihm fo boh sids of h compuio. Th lf hd sid is l = ( l 500 l 400) = , d h igh hd sid is l ( ) Thus, boh qui clos o ch oh. Dwig wo pois of l GD o ul logihmic vicl is gis wo pois i im (s Figu -), w fid h h slop of wo pois

4 4 l 500 l 400 pss h ul gowh of l GD. Mo glly sig, if ul logihm of im-sis d is dw gis im, h h slop cospods o h pcg chg p ui of im...2. Empl 2: Th ul of 70 Th scod pplicio of h ppoimio fomul is bou h ul of 70, which ss h if h of is 00 i % ims h umb of ys fo ivsm is qul o 70, h picipl will doubl. Fo mpl, if you sv 7% is fo ys, you moy will doubl ppoimly; ( ) 0 = If you pu you moy much low of is %, i ks bou 70 ys o doubl; ( + 0.0) 70 = Why dos h ul of 70 hold? Ou ppoimio woks qui wll o pov i. Th ul of 70 c b wi i mhmicl m: if 00i = 70, h ( i) + 2 L us k ul logihm fo boh sids. Fo h lf hd sid, w hv 70 l ( + i) = l ( + i) i = = Fo h igh hd sid, l 2 = Hc, h ul of 70 holds s ppoimio!..3. Empl 3: Th Fish quio L us mov o lil mo complicd mpl. I bliv h you ld h Fish quio i ioducoy mco, o omil is is qul o l is plus pcd iflio. Th Fish quio is pssd s follows: i = +, + + wh i is im- o-y omil is,, + is l is bw d +, is cu pic lvl, d + is h pcio of o-piod hd pic lvl. Now, you hv o hudd y. A l vlu of 00 y is 00. O h oh hd, 00( + i ) you will obi i l m s fui of o-y svig. Th is, you +

5 sv 00 i l m, d pcd o civ 00 + i + y. You svig pl is summizd by h blow bl. 5 i l m f o icipl im Sum wih is im + Nomil m 00 ( + i ) 00 Rl m i + + A l goss is bw d + ( +, ) is compud s h l vlu of picipl d is im + dividd by h l vlu of picipl im, o 00 ( + i ) + + i +, + = = L us ppoim h bov quio. A sy p is o k ul logihm of h fis m; ( ) logihm of h hid m; l +, +, +. A lil mssi p is o k ul + i + l = l ( + i) ( l l + ) i + Combiig wo ppoimd ms lds o quio!, + + = i. Th is h Fish

6 6.2. Cu vibls s ih h sul of h ps o h mio of h fuu A fis-diffc li quio sblishs li lioship bw coscuiv im-sis vibls, sy wh d such s = + z (-2-) is dogously dmid by h bov quio, im-sis vibl, d is pm. Suppos h is bw zo d o, d bov fis diffc quio, w hv = + z 2 2 = + z = 0 + z0 z is ogous ss fom im 0. Giv h Subsiuig hs quios io quio (-2-) i cusiv m, w hv h followig soluio o quio (-2-): 0 (-2-2) = 0 = z + Thus, flcs h ps vlus of z giv 0. Bcus ppochs zo s is lg, mo dis ps z hs wk ffcs o h cu vlu of. L us pick up o mpl fo = + z wh 0< <. Giv physicl cpil K h bgiig of im -, goss ivsm I of dpciio δ wh 0 < δ < is dpciio, coibus o ics i physicl K cpil K K. Thus, fis-diffc li quio holds: K K I δ K = o No h 0< δ <. K = K + I δ Th subsiuio pocdu gs h soluio o h bov quio. ( δ ) 0. (-2-3) K = I + K = 0 Th is, h cu lvl of physicl cpil ivsm I, giv K 0, whil mo dis K flcs h ps vlus of goss I hs wk ffcs o K. H, I would lik you o py cful io o h diffc i h u of im sis bw physicl cpil d goss ivsm. Th is, physicl cpil K svs s sock vibl which cods coomic s poi of im o h bgiig of im, whil goss ivsm I svs s flow vibl which cods coomic civiy fo piod of im o duig im. Thus, quio (-2-3) implis h h cu s pssd by sock vibl flcs h ps

7 7 coomic civiy pssd by flow vibls. W ow suppos h o 0< <, bu < fo = + z. Th, h soluio o his quio, o quio (-2-3) looks h sg; mo dis ps I hs o wk, bu sog ffcs o. Bu, smll gm of quio (-2-) givs us isig sul. = + w (-2-4) + wh ogous vibl w is dfid s z. No h 0< <. Giv quio (-2-4), w hv + = w+ + 2 = w = w+ 3. Rcusivly subsiuig hs quios io quio (-2-4) ifii ims, w div h followig soluio o quio (-2-4). = w+ + lim + (-2-5) 0 = Equio (-2-5) hs h ipio which is opposi o h of quio (-2-2). Th cu vlu of dogous vibl flcs o ps, bu fuu ogous vibls w +. Bcus ppochs zo s is lg, h fis m of h igh hd sid of quio (-2-4) implis h mo dis fuu w + hs wk ffcs o. O h oh hd, is scod m covgs o zo wh is posiiv d fii. Figuivly sig, quio (-2-) implis h h ps siuio dmis h cu s. Covsly, quio (-2-4) suggss h h fuu siuio dmis h cu s. You will s my mpls of quio (-2-4) houghou his cous. I jus pick up o simpl mpl. Suppos h ivsos qui quiy us o b ρ > 0. A cu quiy pic is p, d i is pcd o b p + ogh wih dividd d + i h piod. I his sup, ivsm fui bw im d im + cosiss of cpil gi p + p d icom gi d +. As cosquc p+ p + d+ of quim fom ivsos, quiy u mus b qul o ρ ; p

8 8 p p + d p = ρ >. This codiio ducs o: p = p + d + ρ + ρ + + (-2-6) Equio idd blogs o quio (-2-4) bcus 0< < by cosucio. + ρ Applyig quio (-2-5), h soluio o quio (-2-6) lds o: p = d + lim p + + ρ = + + ρ Giv posiiv fii p, h scod m of h igh hd sid of h bov quio covgs o zo. Th, h cu quiy pic { d, d, d,..., d } wih h of discou ρ. p flcs h fuu dividds You my o wi fo oh mpls of quio (-2-4), bu I would lik you o b lil pi.