Mathematical Notation Math Calculus for Business and Social Science

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1 Mthemticl Nottio Mth Clculus for Busiess d Socil Sciece Use Word or WordPerfect to recrete the followig documets. Ech rticle is worth 10 poits d should e emiled to the istructor t jmes@richld.edu. If ou use Microsoft Works to crete the documets, the ou must prit it out d give it to the istructor s he c't ope those files. Tpe our me t the top of ech documet. Iclude the title s prt of wht ou tpe. The lies roud the title re't tht importt, ut if ou will tpe t the egiig of lie d hit eter, oth Word d WordPerfect will drw lie cross the pge for ou. For epressios or equtios, ou should use the equtio editor i Word or WordPerfect. The istructor used WordPerfect d 14 pt Times New Rom fot with 0.75" mrgis, so the m ot look ectl the sme s our documet. The equtios were creted usig 14 pt fot. For idividul smols (µ, σ, etc) withi the tet of setece, ou c isert smols. I Word, use "Isert / Smol" d choose the Smol fot. For WordPerfect, use Ctrl-W d choose the Greek set. However, it's ofte esier to just use the equtio editor s epressios re usull more comple th just sigle smol. If there is equtio, put oth sides of the equtio ito the sme equtio editor o isted of cretig two ojects. There re istructios o how to use the equtio editor i seprte documet or o the wesite. Be sure to red through the help it provides. There re some emples t the ed tht wlk studets through the more difficult prolems. You will wt to red the hdout o usig the equtio editor if ou hve ot used this softwre efore. If ou fil to tpe our me o the pge, ou will lose 1 poit. Do't tpe the gred out hits or remiders t the ottom of ech pge. These ottios re due t the egiig of clss o the d of the em for tht chpter. Tht is, the chpter 1 ottio is due o the d of the chpter 1 test. Lte work will e ccepted ut will lose 0% of its vlue per clss period.

2 Chpter 1 - Prelimiries Asolute Vlue Rtiol Epoets, 0 =, < 0 m/ m = = + Differece of two squres ( )( ) ± + = ± Perfect squre triomil ( ) Sum/Differece of two cues 3 ± 3 = ( ± )( + ) Emple of fctorig with rtiol epoets 3 3 = / 1/3 ( ) ( = + ) 5 ( ) ( 5 ) / 1/3 ( = + 3 ) ( 5 ) ( ) 3 1/ 1/3 ( ) ( = + ) ( ) 3 3/ 1/ ( 5)( + 3) ( 5 ) + ( 4)( + 3) ( 5 ) 1/3 /3 3/ 1/ ( ) ( ) ( ) ( ) Qudrtic Formul 1/3 /3 ± = 4c Distce Formul d = ( ) + ( ) Slope 1 1 rise m = = = ru Poit Slope form of lie = m( ) use mtri for the rhs

3 Chpter - Fuctios, Limits, d the Derivtive Compositio of fuctios ( f g)( ) f g( ) = Limits re wht seprte clculus from lger. A limit is ot cocered with wht hppes t prticulr vlue, ol wht hppes s ou pproch tht vlue. Limits c usull e evluted sustitutig the vlue ito the epressio uless tht cuses prolem like divisio zero. If there is prolem, the we tr to mipulte the epressio to elimite the prolem. Emple of idetermite form. The origil form is 0/0 d so we rtiolize the umertor multiplig 1 i the form of the cojugte over itself d simplif. Notice tht we cotiue to write the limit i frot util tht poit where we ctull plug i the vlue of lim = lim ( + ) = lim = lim ( )( + + ) ( )( + + ) = lim = = = A fuctio is cotiuous t poit if the fuctio is defied t tht poit, the limit s ou pproch tht poit eists, d the vlue of the fuctio is equl to the limit. Averge rte of chge Istteous rte of chge m m sec t ( + ) ( ) f h f = h f + h f = lim h 0 h ( ) ( )

4 Chpter 3 - Differetitio Power Rule Product Rule ( ) d 1 = d fg = fg + f g Quotiet Rule f gf fg = g g d f g f g g d = d d du = d du d Chi Rule ( ) ( ) ( ) Chi Rule I Eglish, the Chi Rule ss to tke the derivtive of the outside fuctio times the derivtive of the iside fuctio. I ecoomics, "mrgil" mes derivtive. So mrgil cost is, d mrgil profit is. R P Averge Cost C( ) Elsticit of Demd E( p) C( ) = pf = f ( p) ( p) C, mrgil reveue is Demd is elstic if elsticit is greter th 1, uitr if elsticit is 1, d ielstic if elsticit is less th 1. A differetil is chge i vrile. A derivtive is rte of chge d is the rtio of two differetils. So d d d re differetils d d/d is derivtive. d ( ) d= f. Locl Lier Approimtio ( ) ( ) ( ) ( ) d = f = f + f + f

5 Chpter 4 - Applictios of the Derivtive A fuctio is icresig where the first derivtive is positive d decresig where the first derivtive is egtive. A criticl poit is vlue i the domi of the fuctio where the first derivtive is either zero or udefied. A fuctio is cocve up whe the secod derivtive is positive d cocve dow where the secod derivtive is egtive. A iflectio poit is poit where the cocvit chges. A cotiuous fuctio hs reltive mimum t criticl poit if the fuctio is icresig to the left d decresig to the right of tht poit. A cotiuous fuctio hs reltive miimum t criticl poit if the fuctio is decresig to the left d icresig to the right of tht poit. A cotiuous fuctio hs reltive mimum t criticl poit if the secod derivtive is egtive t tht poit d reltive miimum t criticl poit if the secod derivtive is positive t tht poit. Verticl smptotes of rtiol fuctio occur whe the deomitor is zero. Horizotl smptotes re foud tkig the limit s. ± A cotiuous fuctio o closed itervl will hve oth solute mimum d solute miimum. The grph of the fuctio 16/ = is show to the right. Notice tht where hs reltive miimum, ' crosses the -is d where there is iflectio poit, " crosses the -is. Use wiplot to crete the grph

6 Chpter 5 -Epoetil d Logrithmic Fuctios = = log The grph of d o the sme grph. Notice the re reflectios of ech other out the lie = (iverses of ech other). The epoetil fuctio hs domi of ll rel umers d rge of while the logrithmic fuctio hs domi of rel umers. Use wiplot to crete grph > 0 > 0 d rge of ll 1 e= lim 1+ = lim 1+ 0 Coversio etwee forms = log = Log of product log = log + log Log of quotiet log log log = Defiitio of e ( ) 1/ Log with power log = log Compoud Iterest 1 A= P + r m r Effective Rte of Iterest reff = 1+ 1 m m mt Derivtive of epoetil Derivtive of logrithm d e e d = d 1 l d =

7 Chpter 6 - Itegrtio Sice the derivtive of costt is 0, evidece of the costt tht eisted i the origil fuctio is lost whe the derivtive is foud. Whe fidig the tiderivitive, we compeste for this ddig costt. This shows up s +C t the ed of ever idefiite itegrl. The vlue of C c e foud for iitil vlue prolems. Power Rule for Itegrls Itegrl of epoetil Itegrl of reciprocl ed= e + C + 1 d= + C 1 d = l d + C Itegrtio sustitutio is the tiderivtive form of the chi rule. The defiite itegrl is equl to the re uder curve. This is foud tkig limit of sum s the su-itervls get smller d smller. Defiite Itegrl ( ) lim ( ) f d= f k k= 1 Tht defiitio mkes sese, ut it's pi to work with. More useful is the Fudmetl Theorem of Clculus, which sttes tht if F is tiderivtive of cotiuous fuctio ( ) = ( ) ( ) f, the. f d F F 1 f = ve f d Averge Vlue ( ) Cosumer's Surplus ( ) CS = D d p Producer's Surplus ( ) 0 PS p S d = 1 Coefficiet of Iequlit = ( ) L f d 0 0

8 Chpter 7 - Additiol Topics i Itegrtio Itegrtio prts is sed is developed from the product rule for derivtives. udv= uv vdu. You eed to divide the origil itegrl ito two prts, u d dv. You must e le to itegrte dv d du should e simpler th u. A crom to help ou choose u is LAE, which stds for Logrithm fuctios, Algeric fuctios, d Epoetil fuctios where ou let u e the first kid ou ecouter. This does't lws work, ut it will work much of the time. A tle of itegrls provides quick w to evlute some itegrls. You m eed to mke sustitutios efore lookig up the vlue i the tle. A short tle of itegrls is foud i sectio 7. of our tet. Sometimes defiite itegrls just c't e evluted lgericll. I these cses, we hve umericl methods like the trpezoid rule or Simpso's rule tht c e used to pproimte the defiite itegrl. Trpezoid Rule. The vlue of ech iterior poit is couted twice. f d ( ) = [ ] Simpso's Rule, umer of itervls must e eve. The vlues of ech iterior poit lterte etwee eig couted four times d eig couted twice. f d 3 ( ) = [ ] 0 1 1

9 Chpter 8 - Clculus of Severl Vriles To grph fuctio of two vriles, (, ) z = f, we c either grph three- dimesiol surfce or two-dimesiol cotour mp mde up of level curves. I cotour plot, the closer the level curves re together, the more rpidl the vlue of the fuctio is chgig. A prtil derivtive is rte of chge with respect to oe of the vriles while the other vriles re held costt. f = f f = f is the first prtil derivtive of f with respect to d is the first prtil derivtive of f with respect to. A criticl poit is poit i the domi where either oth first prtil derivtives re zero or t lest oe of the first prtil derivtives does ot eist. Assumig criticl poit is the kid where oth first prtil derivtives re zero, the ou c determie whether the vlue is mimum or miimum usig the secod derivtive test. D= f f f f = f f f Secod Derivtive Test If D > 0 d f < 0, the the criticl poit is reltive mimum. If D > 0 d f > 0, the the criticl poit is reltive miimum. If, the the criticl poit is sddle poit d if, the test is icoclusive. D < 0 D = 0 Lgrge Multipliers. To fid the reltive etrem of fuctio f suject to costrit uilir fuctio (,, λ) = (, ) + λ (, ) F f g ( ) g, = 0. The solve the sstem of equtios tht results settig ech prtil derivtive equl to zero. Tht is, simulteousl set,, d. F = 0 F = 0 F λ = 0, form