IMP007 Introductory Math Cours 3. ARITHMETICS AND FUNCTIONS 3.. BASIC ARITHMETICS REVIEW (from GRE) Which numbrs form th st of th Intgrs? What is th product of an intgr multiplid by zro? and dividd by zro? What is th sign of th multiplication of two intgrs of diffrnt sign? and of th sam sign? What is th rsult of 5? 3 What is th rsult of 7 3? 8 5 What is th rsult of 6.5 + 68.3? /7
IMP007 Introductory Math Cours How much is 5.38 0.4? What is th rsult of m 5, m 0, m, m -3, m /? m 5 m 0 m m m m m m m m 3 3 m m / m What is th rsult of (-4), 4, 0, 5 0, 6 6 0 9? 4 /7
IMP007 Introductory Math Cours If a quantity incrass from 600 to 750, what prcntag of incras is this? And if it dcrass from 500 to 400, what is th prcnt dcras? What is th absolut valu of ½, 0, -.6? What is an algbraic prssion? What is a variabl? A variabl is somthing whos magnitud can chang, i.. somthing that can tak diffrnt valus. Variabls frquntly usd in conomics includ pric, profit, rvnu, cost, national incom, and so on. Sinc ach variabl can assum various valus it must b rprsntd by a symbol instad of a spcific numbr. For ampl, w may rprsnt pric by P, profit by, rvnu by R, cost by C, national incom by Y, and so forth. 3/7
IMP007 Introductory Math Cours Which ar th two typs of variabls common in conomic modls? Endognous and ognous variabls. Endognous variabls (originating from within) ar variabls whos solution valus w sk from th modl. Eognous variabls (originating from without) ar variabls which ar assumd to b dtrmind by forcs trnal to th modl, and whos magnituds ar accptd as givn data only. Th sam variabl can b ognous som tims, and othr tims ndognous, it will dpnd on th assumptions of th modl. What is a constant? A constant is a magnitud that dos not chang and is thrfor th antithsis of a variabl. What is a cofficint? A constant that is joind to a variabl. For instanc 7, 7 is th cofficint and th variabl. What is a paramtr? A cofficint that is symbolic rathr than numrical. W do not assign to it any spcific numbr, and thrfor it can tak virtually any valu. In short, a paramtr is a constant that is variabl! For ampl, ap, whr a is th paramtr and P th variabl (likly pric). In th prssion 7 5, what ar th cofficints, what is th constant? How could you simplify th following prssions:, 3y y 3? 5, 3 6 4/7
IMP007 Introductory Math Cours How could you simplify th following prssions 4, 5y 9y, 7 4 4? How could you simplify th following prssion 9? 4 What is th rsult of 3 7? 5/7
IMP007 Introductory Math Cours What is an quation? Which ons ar th thr typs of quations w may distinguish in conomics? Dfinitional quations, bhavioural quations, and quilibrium conditions. Dfinitional quations st up an idntity btwn two altrnat prssions that hav actly th sam maning. For such an quation, th idntical-quality sign (rad: is idntically qual to ) is oftn mployd in plac of th rgular quals sign =, although th lattr is also accptabl. As an ampl, total profit is dfind as th css of total rvnu ovr total cost; w can thrfor writ R C. Bhavioural quations, spcify th mannr in which a variabl bhavs in rspons to changs in othr variabls. This may involv ithr human bhaviour (such as aggrgat bhaviour) or nonhuman bhaviour (such as how total cost racts to output changs). For instanc, C 75 0Q. Equilibrium conditions, hav rlvanc only if th modl involvs th notion of quilibrium. If so, th quilibrium condition is an quation that dscribs th prrquisit for th attainmnt of th quilibrium. For instanc, Q Q [quantity dmandd = quantity supplid] d s 6/7
IMP007 Introductory Math Cours How could b th following prssions simplifid? a a a a, ( )( y ), b,, y a a b, ( )( ) a b ( ), 0 3 How could w simplify th prssion? 3/ How many mthods can you us to solv th following systm of quations? Could you solv it using thm? 3 7/7
IMP007 Introductory Math Cours What is a quadratic prssion? which formula would you us to solv it? How many solutions hav quadratic prssions? 8/7
IMP007 Introductory Math Cours Th solutions of th quadratic prssions that ar ral numbrs ar usually calld roots. Thr will b two ral roots if If b 4ac. b 4ac thn th two ral roots will coincid and only on ral valu of will satisfy th quation. If b 4ac thn thr will b no ral root. Somtims th trm discriminant is usd. Th discriminant is dfind as b 4ac. Thrfor if 0 thr is only on root. If 0 thr ar no ral roots, and if 0 thr will b two roots. Solv th quadratic quation 6 0. Could you solv in a diffrnt way th quadratic quation 6? 9/7
IMP007 Introductory Math Cours 0/7 Th factoring tchniqu is basd on th fact that som quadratic prssions can b statd as th product of two factors, for ampl: d d d d d d d d d Could you solv by factoring th quadratic prssion 5 8? What is an inquality? How ar inqualitis solvd?
IMP007 Introductory Math Cours 4 9 Could you solv th inquality 5? Could you solv th following prssions 4 i i y whr i, i,,3,... i? and 4 y i? i y=++3+4=0 y=34=4 /7
IMP007 Introductory Math Cours BIBLIOGRAPHY http://www.ts.org/mdia/tsts/gre/pdf/gremathpractic.pdf Alpha C. Chiang (984) Fundamntal Mthods of Mathmatical Economics Third dition. McGraw-Hill, Inc. Ch. 3..FUNCTIONS 3.. Th cartsian spac W saw in st thory th dfinition of Cartsian Product. W can us it to gnrat ordrd pairs. W call ordrd pairs to thos pairs, dnotd as {a,b} whr th ordr of th lmnts mattr, that is {a,b} {b,a} unlss a=b. In opposition, unordrd pairs ar thos pairs for which {a,b}={b,a}. For sts with mor lmnts w can spak of ordrd or unordrd sts. On important cas is whn w apply cartsian product multiplying th st of th ral numbrs: X Y, y R, y R whr R is th st of th ral numbrs. Each ordrd pair of ral numbrs is calld a coordinat and corrsponds to a uniqu point in th cartsian spac. Th cartsian spac is dfind by two ais rprsnting th ral numbrs: Th horizontal ais calld abscissa contains th. Th vrtical ais calld ordinat and contains th y. Th point whr th two ais cross rprsnts th pair (0,0) and is calld th origin and prssd by O. W can prss this spac as th R R or simply R. If th sts, y and z consist of all th ral numbrs, th cartsian product will corrspond to th st of all points in a thr dimnsional spac. This may b dnotd by R R R, or mor simply R 3. If cartsian product is applid to n sts consisting of all th ral numbrs, th rsult will corrspond to th n-dimnsional spac, prssd by R n. /7
IMP007 Introductory Math Cours 3... Rlation Sinc any ordrd pair associats a y valu with a valu, any collction of ordrd pairs will constitut a rlation btwn y and. For instanc th sts, y) y (, y) y ( and A rlation may b such that for ach valu thr ists only on corrsponding y valu. Thn y is said to b a function of, and it is dnotd by y=f(). A function is thrfor a st of ordrd pairs with th proprty that any valu uniquly dtrmins a y valu. Th customary symbols for rprsnting a function ar: f, g, F, G, and th Grk lttrs, and thir capitals,. In th function y=f(), is rfrrd as th argumnt of th function, and y is calld th valu of th function. Th domain of a function is th st of all prmissibl valus that can tak in a givn contt. Th st of all valus that th y variabl can tak is calld th rang of th function. It is also typical to start by dfining a function s domain and rang using th following notation: for th cas of a function with domain in th natural numbrs and rang in th ral numbrs. It is also common to rfr to as th indpndnt variabl and to y as th dpndnt variabl. Ercis 3/7
IMP007 Introductory Math Cours 3..3. TYPES OF FUNCTIONS W dfind a function as y=f() but f() can tak diffrnt forms. Constant functions A function whos rang consist only of on lmnt. For ampl, y=f()=7 Eognous variabls ar typically constant. Polynomial functions Th word polynomial mans multi-trm. A polynomial function of a singl variabl has th gnral form y a a a... 0 n a n Dpnding on th valu of n w hav diffrnt typs of polynomial function: Cas of n=0: y=a 0 Constant function Cas of n=: y=a 0 +a Linar function Cas of n=: y=a 0 +a +a Quadratic function Cas of n=3: y=a 0 +a +a +a 3 3 Cubic function Th valu of n is calld th dgr of th polynomial. Rational functions A rational function is a function in which y is prssd as th ratio of two polynomials in th variabl. For ampl y Non-algbraic functions 4 A rational function is a function in which y is prssd in trms of polynomials and / or roots (such as squar root) of th variabl. For ampl y 3 Som important non-algbraic functions ar: ponntial functions y=b 4/7
IMP007 Introductory Math Cours logarithmic functions y=log b trigonomtric functions Functions with mor than on indpndnt variabl For instanc z=g(,y). Now th domain will not b, or anothr st of numbrs, but a st of ordrd pairs (,y). This function g will map a point in a two-dimnsional spac into a point on a lin sgmnt. This function thus, will gnrat ordrd tripls (,y,z). A typical ampl is th common prssion of output Y as a function of labour L and capital K, or Y=Y(L,K) Whn many variabls ar involvd in a function, w dnot th variabls with,...,, n. Solving highr-dgr polynomial functions W know how to solv quadratic functions. What about cubic or or quartic functions? Th trick is to apply factoring in th appropriat way. Givn an n th dgr polynomial function f(), w can pct n roots. First, try to find a constant c such that th function is divisibl by th factor (+c ). Th quotint f()/(+c ), lt s call it g(), will b a polynomial function of dgr (n-). It follows that f ( ) ( c ) g( ) Rpating th procss, it will possibl to rduc th original n th dgr polynomial f() to a product of actly n trms: f ( ) ( c ) ( c ) ( cn In ordr for f() to b qual to zro, at last on of th factors must b qual to zro, thrfor w can find n roots: ( i,,, n) i c i Th problm is to find th factors. You can only us your intuition or a tdious trial and rror procss. ) 5/7
IMP007 Introductory Math Cours 3..4. GRAPHIC REPRESENTATION OF FUNCTIONS Th graphic rprsntation of som of th most typical functions is as follows: S Alpha Chiang pag 6. You can also chck th graphic rprsntation of a function at http://rchnronlin.d/function-graphs/ 6/7
IMP007 Introductory Math Cours 3.3. EXPONENTIAL AND LOGARITHMICAL FUNCTIONS 3.3. Eponntial functions A function whos indpndnt variabl appars in th rol of an ponnt is calld an ponntial function. For ampl y=f(t)=b t whr t is th variabl and b a constant calld th bas of th ponnt. Bas convrsion: it is th procdur to chang th bass and th ponnts. It only works if y can b prssd as powrs of various altrnativ bass. For instanc, y t t t 9 (3 ) 3. By this procdur, all bass that ar 0<b<, can b transformd into bass b >. For ampl, 0. t t t t (5 ) 5 5 Warning: t t b (b)!!! GRAPHIC REPRESENTATION OF LOGARITHMS Natural ponntial functions In conomics you will oftn find functions lik this: y=, that is, ponntial functions with bas. What kind of a numbr is? 7/7
IMP007 Introductory Math Cours lim f ( m) lim m m m m.788... Th numbr =.788 can b intrprtd as th yar-nd valu to which a principal of will grow if intrst at th rat of 00 prcnt pr annum is compoundd continuously. If w allow for mor yars of compounding, assum t priods, thn th final valu V (m) will b t.... If furthrmor w allow for principals of unspcifid amount of A, thn at th nd of priod t th V(m) will b t A. And allowing for intrst rats othr than 00 prcnt, say nominal intrst rat is r, thn it can b shown that V=A rt. So th formula V=A rt is conomically intrprtd as th continuous intrst compounding. Instantanous rat of growth Apart from an intrst rat, this formula can also b applid to procsss of ponntial growth, lik population, walth, or capital. Now r rprsnts th instantanous rat of growth of th function V=A rt. Instantanous mans that it is calculatd at a particular instant, at a momnt t. Hr growth rat is constant, but it could also chang according to t, for instanc r(t)= (t) Discrt growth. Procsss of growth at discrt compounding intrst rat dtrmin th final valu of th principal according to th formula V ) t A( i. But for this function w can always find an r such that ( i) t A rt Discounting and Ngativ growth 8/7
IMP007 Introductory Math Cours Th problm of discounting is finding th prsnt valu A from a futur valu V. V t So if V A( i), now A V ( i t ( i) ) t for th discrt cas. For th V rt continuous cas: A V. rt Discounting can also prss ngativ growth, bing r known calld somtims th rat of dcay. EXERCISE 3.3.. Logarithmic functions LOGARITHMS A logarithm to bas b of a numbr is an ponnt y that satisfis =b y. It is writtn log b () or, if th bas is unambiguous, as log(). So th logarithm is th powr to which a bas must b raisd to attain a particular numbr. In othr words is quivalnt to Th bas b must b nithr 0 nor and is typically 0,, or. Whn and b ar rstrictd to positiv ral numbrs thn log b () is a uniqu ral numbr. For ampl Warning: a ngativ numbr or zro cannot posss a logarithm!!! Common logarithm: in computational work it is oftn usd th logarithm of bas 0, calld common logarithm or log 0 (or if th contt is clar, simply log) 9/7
IMP007 Introductory Math Cours Natural logarithm: (for natural log) th most usual in conomics. It is th logarithm with bas. Also calld ln() and somtims it is also rfrrd as log() if it is not ambiguous from th contt. For instanc, y t t log y (or t ln y ) ln ln 3 log log ln log 0 ln log 3 3 0 In gnral: ln n n Ruls of logarithms Th following ruls ar trmly usful to simplify mathmatical oprations, spcially ponntial functions. Rul I log of a product: ln( ab) ln a ln b 6 4 6 4 Eampl : ln( ) ln ln 6 4 0 7 7 Eampl : ln( A ) ln A ln ln A 7 Warning!!! ln( a b) ln a ln b and ln( a b) ln a ln b. a b Rul II log of a quotint: ln ln a ln b Eampl 3: ln ln ln c ln c c 5 Eampl 4: ln 5 ln ln 5 3 Rul III log of a powr: ln u a a ln u Eampl 5: ln 5 5ln 5 0/7
IMP007 Introductory Math Cours Eampl 5: ln A 3 3ln A Logarithmic functions Whn a variabl is prssd as a function of th logarithm of anothr variabl, th function is rfrrd to as logarithmic function. For instanc, t log b y or t log y ln y Logs ar usd to transform non-linar quations into quations that ar linar in th cofficints. For ampl y=a b, can b transformd by taking logs on both sids into: log y log a b log which is linar in th cofficints. EXERCISE 3 Bibliography on ponntial and logarithmic functions: Alpha C. Chiang (984) Fundamntal Mthods of Mathmatical Economics Third dition. McGraw-Hill, Inc. Ch.0 3.4. LIMITS Informally, th concpt of limit is concrnd with th qustion: What valu dos on variabl (say, q) approach as anothr variabl (say v) approachs a spcific valu (say, zro)? Or if w hav a function y=f() finding th limit is quivalnt to answring y tnds to L, as tnds to a or lim y L a It can b that whn approachs a from th lft th limit is diffrnt than whn approachs a from th right. This is to say that th lft-sid limit of y whn a is diffrnt than th right-sid limit of y whn a. Lft-sid limit of y whn a is rprsntd by lim y. a /7
IMP007 Introductory Math Cours Right-sid limit of y whn On ampl of this is lim 0 a is rprsntd by lim y. y, givn that y Asymptot: An asymptot is a straight lin or curv A to which anothr curv B (th on bing studid) approachs closr and closr as on movs along it. As on movs along B, th distanc btwn it and th asymptot A tnds to bcom smallr and smallr ovrall, and vntually nvr bcoms longr than a any spcifid distanc. A curv may or may not touch or cross its asymptot. If a curv C has th curv L as an asymptot, on says that C is asymptotic to L. Plas not that is not a numbr. Graphical ampl: function q=g(v): /7
IMP007 Introductory Math Cours Figur a, th limit is th sam from both sids. Thrfor w can writ lim q L vn Figur b, th curv is not smooth, thr is a turning point at point N. Nvrthlss, th limit is th sam from both sids lim q L. vn Figur c, shows a stp function. Th right sid limit at N is L and th lftsid limit is L. Hnc, q dos not hav a limit as v N. Figur d: th right-sid limit at N is +, approaching th brokn vrtical lin as an asymptot, whras th lft-sid limit is - also approaching also th brokn vrtical asymptot. Th lim q dos not ist. On th vn othr hand, anothr intrsting limit ist in th graph: lim v q M and lim q M. v How to valuat th limit of a function at a point. On has to find th valus of y for diffrnt valus of approaching incrasingly th point to b valuatd, both from th right and from th lft. Normally, th limit at on point will coincid with th valu of a function at that point (s latr continuity). For ampl, if w hav to find for th function y=. lim Th valu of y at = is, and you can chck it, for ampl, with f(.9), f(.99) and f(.999) from th lft, and f(.), f(.0) and f(.00) from th right. It is clar that both from both sids, th valus of y approach. f(.9) f(.99) f(.999) f() f(.00) f(.0) f(.).9.99.999.00.0. y Limit at a point not blonging to th domain or not dfind: 3/7
IMP007 Introductory Math Cours Cas 0 0 : Considr th cas whr ƒ() is undfind at = c. In this cas, as approachs, f() is undfind at = but th limit quals : f(0.9) f(0.99) f(0.999) f(.0) f(.00) f(.0) f(.).95.99.999 undf.00.00.0 Thus, f() can b mad arbitrarily clos to th limit of just by making sufficintly clos nough to. Th indtrminat forms ar: Thr ar many diffrnt mthods to find th limit at an undfind point. On possibility is to find an quivalnt prssion that is not undfind at that point. For instanc, givn q=(-v )/(-v), find lim v q. Hr N= is not in th domain of th function, it is indtrminat of th form 0/0. Howvr, w know v v v that q. Som othr usful tricks : k a) Indtrminat of typ, k 0 0, thrfor q ( v) for v. Whn v, it is clar In ordr to solv thm, calculat th sid limits. If thy ar qual, th function has a limit. If thy ar diffrnt, th limit dos not ist. 4/7
IMP007 Introductory Math Cours Eampl lim lim 0 lim Th sid limits ar diffrnt. Thrfor th limit dos not ist. b) Indtrminat of typ 0 0 This typ of indtrminat disappars by factoring th numrator and th dnominator and simplifying. Eampl 3 lim 0 0 ( )( ) lim c) Indtrminat of typ lim( ) 3 This typ of indtrminat disappars by dividing numrator and dnominator by th maimum powr of th dnominator. Eampl 4 lim (4 ) / lim ( ) / 4 lim 4 Anothr possibility is to apply l'hôpital's rul. For compl functions it is convnint to us th limit thorms: Thorm I if q=av+b, thn lim q an b (a and b ar constants). vn 5/7
IMP007 Introductory Math Cours Eampl Givn q=5v+7, w hav lim q 5 7 7 v Thorm II if q=g(v)=b, thn lim q b. That is, th limit of a constant is a constant. vn Thorm III if q=v, thn lim q N. vn if q=v k, thn lim q vn N k Thorms involving two functions If w hav two functions of th sam indpndnt variabl v, q =g(v) and q =h(v), and if both functions posss limits as follows: lim q vn L and lim q L vn whr L and L ar two finit numbrs, th following thorms ar applicabl: Thorm IV sum-diffrnc limit thorm lim( q vn q) L L In particular, lim( q q) L vn Thorm V product limit thorm lim( q vn q) L L In particular, lim( q vn q ) L Thorm VI quotint limit thorm lim( q vn / q) L / Limit of a polynomial function L 6/7
IMP007 Introductory Math Cours n Givn th polynomial function q g( v) a a v a v... a v 0 n, th limit as v tnds to N will b, according to th prvious thorms: lim q vn g( v) a 0 a N a N... a n N n EXERCISE 6 CONTINUITY Whn a function q=g(v) posssss a limit as v tnds to th point N in th domain, and whn this limit is also qual to g(n) - that is, qual to th valu of th function at v=n th function is said to b continuous at N. Bibliography on limits: Alpha C. Chiang (984) Fundamntal Mthods of Mathmatical Economics Third dition. McGraw-Hill, Inc. pgs. 3-47 7/7