STUDY COURSE BACHELOR OF BUSINESS ADMINISTRATION (B.A.)

STUDY COURSE BACHELOR OF BUSINESS ADMINISTRATION (B.A. MATHEMATICS (ENGLISH & GERMAN REPETITORIUM 0/06 Prof. Dr. Philipp E. Zeh Mthemtis Prof. Dr. Philipp E. Zeh LITERATURE (GERMAN Böker, F., Formelsmmlug für Wirtshftswisseshftler. Mthemtik ud Sttistik, Mühe 009. Böker, F., Mthemtik für Wirtshftswisseshftler. Ds Üugsuh,. Auflge, Mühe 0. Gehrke, J. P., Mthemtik im Studium. Ei Brükekurs,. Auflge, Mühe 0. Hss, O., Fikel, N., Aufge zur Mthemtik für Wirtshftswisseshftler,. Auflge, Mühe 0. Sdseter, K., Hmmod, P., Mthemtik für Wirtshftswisseshftler. Bsiswisse mit Prisezug,. Auflge, Mühe 0. Thoms, G. B., Weir, M. D., Hss, J. R., Bsisuh Alsis,. Auflge, Mühe 0. Mthemtis Prof. Dr. Philipp E. Zeh

LITERATURE (ENGLISH Chig, A. C., Wiwright, K., Fudmetl Methods of Mthemtil Eoomis, rd Editio, Bosto 00. (MGrwHill Dowlig, E. T., Shum's Outlie of Mthemtil Methods for Busiess d Eoomis, Bosto 009 (MGrwHill. Sdseter, K., Hmmod, P., Essetil Mthemtis for Eoomi Alsis, th Editio, Hrlow et l 0. (Perso Tlor, R., Hwkis, S., Mthemtis for Eoomis d Busiess, Bosto 008. (MGrwHill Zim, P., Brow, R. L., Shum's Outlie of Mthemtis of Fie, d Editio, New York et l 0 (MGrwHill. Mthemtis Prof. Dr. Philipp E. Zeh REPETITORIUM - TOPICS. Itrodutor Topis.. Numers.. Alger.. Sequees, Series, Limits.. Polomils. Lier Alger.. Sstem of Lier Equtios.. Sstem of Lier Iequlities. Differetil Clulus.. Bsis.. Derivtive Rules.. Applitios & Eerises - Curve Skethig. Itegrl Clulus.. Bsis.. Rules for Itegrtio.. Applitios & Eerises Mthemtis Prof. Dr. Philipp E. Zeh

. Itrodutor Topis Mthemtis Prof. Dr. Philipp E. Zeh.. NUMBERS 6 0 turlumersi.e {,,,...} 6 0 itegersi.e Z {..., -,-,-,-,0,,,,,...} 0 Q rtiolumers( Z d frtios, 0 IR relumers( Q dirrtiolumers, roots,, e... 0 C ompleumers(ir dimgirumers - i where i - Mthemtis Prof. Dr. Philipp E. Zeh 6

.. ALGEBRA.. Lws Commuttive lw: + = + = Assoitive lw: ( + + = + ( + ( = ( Distriutive lw: ( + = +.. Biomil Theorems. Biomil theorem: ( + = + +. Biomil theorem: ( - = - +. Biomil theorem: ( + ( - = - Mthemtis Prof. Dr. Philipp E. Zeh 7.. ALGEBRA.. POWERS Are of squre : Volume of ue : A hh V hhh Shortut A h V h Geerl... with ftors Well: sis epoet Emples: z z z z Mthemtis Prof. Dr. Philipp E. Zeh 8

Mthemtis Prof. Dr. Philipp E. Zeh 9.. ALGEBRA.. POWERS CALCULATION RULES FOR EXPONENTS N, M m m m m m m m : 6 d d d d d d d d d Emple: Emple: Emple: Mthemtis Prof. Dr. Philipp E. Zeh 0.. ALGEBRA.. POWERS Emples: 6 6 7 7 0 6 0 0 ; ; 0. 0. 0 :0 0 ; : ; u :u u 0 0 0 ; ;

.. ALGEBRA.. Roots Rules for positive, d m, speil se : : m m m m m m ; m SPECIAL CASE: NEGATIVE RADICAND omple umer (ot overed here Mthemtis Prof. Dr. Philipp E. Zeh.. ALGEBRA.. Roots, sie 6 6 6 6, sie 6 6 8,sie 8, sie 0, sie 0 Mthemtis Prof. Dr. Philipp E. Zeh

.. ALGEBRA.. Logrithms Potetil futio,if lookig for e.g 8 Squre root futio,if lookig for e.g 8 Logrithm log,if lookig for epoet equlslogrithmof to thesis Questio: To whih power do ou hve to rise, i order to get s the result Remrk: This power whih hs to e rised i order to get is lled the logrithm of whe is the sis. Mthemtis Prof. Dr. Philipp E. Zeh.. ALGEBRA.. Logrithms Emples: where 6,, 6 6 ow 6,, 6 6 log log 6 6 Oservtio: log is lws tke i orde to some se. i.e we oti smeumer 6 tkig differet se d risig it with differet power. Mthemtis Prof. Dr. Philipp E. Zeh

.. ALGEBRA.. Logrithms More Emples: 0 0 0.0 000 0. 0 0 0.0 000 0. log log 0 log 0 0.0 000 0. Smll Eerise: Idetif, d i ove emples. Tke help from the previous slides. Mthemtis Prof. Dr. Philipp E. Zeh.. ALGEBRA.. Logrithms (speil ses Sie Sie 0 log 0 log Reiprol vlue : log log Mthemtis Prof. Dr. Philipp E. Zeh 6

.. ALGEBRA.. Logrithms Now Cosider: log. multipl m m ( ( d d log ( log ( & ( we get m m log m log m sustitute ( & ( i ( we get log ( from the defiitio of log we hve ( ( This is kow s the first rule/lw of logrithm. Similrl, we lso hve other rules s preseted i the et slide. Mthemtis Prof. Dr. Philipp E. Zeh 7.. ALGEBRA.. Logrithms Rules for logrithms: log log log log log log log log m log log m log log with sis log Mthemtis Prof. Dr. Philipp E. Zeh 8

.. ALGEBRA.. Logrithms Further Proofs: log log log Rell Alsotkelogoothsidesof log log with sis From ( d(,we get log log log log log log log log ( withse ( Mthemtis Prof. Dr. Philipp E. Zeh 9.. ALGEBRA.. Logrithms Proof: log Relllog log (. log log log log times log log log the we hve log log log log log log... log log log... log log log (. log log log (. Mthemtis Prof. Dr. Philipp E. Zeh 0

.. ALGEBRA.. Logrithms Note tht, we develop ssoitios etwee power rules d lws of logrithm: log log log. log log log Mthemtis Prof. Dr. Philipp E. Zeh.. ALGEBRA.. Logrithms Emples: log 8 log log log log log 6 log log log 6 log log 6 log 6 log log log log 6 Smll Eerise: Compute the vlues of ove epressios. Mthemtis Prof. Dr. Philipp E. Zeh

.. ALGEBRA.. Logrithms Emples: log d log log log log log ( log d log.log log d.log d log log ( log z Eerise : Solve log d Mthemtis Prof. Dr. Philipp E. Zeh.. ALGEBRA.. Logrithms Emple: log ( l og log log log log log log Emple: log.. log Oservtio: I the left emple ove, o se is speified for the log. I suh ses, we ssume umer to e the se. However, tpil ovetio is to ssume =0. Mthemtis Prof. Dr. Philipp E. Zeh

.. ALGEBRA.. Logrithms Emple:.. Let ( ( 0 or or log log or log 6. 6. 6 0 or log Mthemtis Prof. Dr. Philipp E. Zeh.. ALGEBRA.. Logrithms log wekow tkeothsidesipowerof log log log i.e. tkigothsidesipowerisatilog Emple: log log log Mthemtis Prof. Dr. Philipp E. Zeh 6

.. ALGEBRA.. Logrithms Speil logrithms: e Nturl logrithm: log e = l = = e (e =,78 l Commo logrithm: log 0 = lg = = 0 Mthemtis Prof. Dr. Philipp E. Zeh 7.. ALGEBRA..6 Ftoril, Asolute Vlue, Sums! (ftoril! = Emple:! = =! = Defiitio: 0! = Asolute vlue: 0 0 0 is lws positive or 0 Emple: 9 9 9 9 0 0 0 Therefore, the solute vlue (or modulus of rel umer is ol the umeril vlue of without regrd to its sig. Mthemtis Prof. Dr. Philipp E. Zeh 8

Mthemtis Prof. Dr. Philipp E. Zeh 9.. ALGEBRA..6 Ftoril, Asolute Vlue, Sums Sigm sig: Emples: 6 0 6 0 0 0... i i i i i i i i i i i i Mthemtis Prof. Dr. Philipp E. Zeh 0.. SEQUENCES, SERIES AND LIMITS Geerl Nottio of Sequee (Folge:...,,, 0,,...,,,.....,, k k e g k k k

.. SEQUENCES, SERIES AND LIMITS A sequee is lled overget, if it hs limit g; i.e., if there is limit g with: lim k k g Emple : Coverget Sequees k k,,,... lim 0 k k (zero sequee k k 0,,,,... lim k k k k k 9 6,,,... 7 lim k k k,7888... e (Euler's umer Mthemtis Prof. Dr. Philipp E. Zeh.. SEQUENCES, SERIES AND LIMITS A sequeeis lled divergetif olimit eists. Emples : DivergetSequees k k,,,... k k,, 6, 8,... Osilltig k Sequee :,,,,,... Mthemtis Prof. Dr. Philipp E. Zeh

.. SEQUENCES, SERIES AND LIMITS Arithmeti Sequees Defiitio: A sequee is lled rithmeti if the followig is vlid: k+ k = d, with d = ostt (for ll elememets of the sequee k =,, Compositio lw for rithmeti sequees: k = + (k - d Emple: Cosider the sequee, 0,, 0... =, = 0, d = - = 0 - = = + (-. = + 0 = Mthemtis Prof. Dr. Philipp E. Zeh.. SEQUENCES, SERIES AND LIMITS Geometri Sequees Defiitio: A sequee is lled geometri, if the followig is vlid: k k q with q = ostt (for ll elemets of the sequee k =,, Compositio lw for geometri sequees: k = k-. q = k-. q = k-. q =.. =. q (k- Emple: Cosider the sequee,, 8, 6... =, =, q = / = =. q = 6. = or. q =. =.6 = Mthemtis Prof. Dr. Philipp E. Zeh

.. SEQUENCES, SERIES AND LIMITS Series (Reihe Durig the ompositio of (ifiite series ll elemets of sequee re summed up: k k The sum up to the -th elemet is lled -th prtil sum: R S k k Mthemtis Prof. Dr. Philipp E. Zeh.. SEQUENCES, SERIES AND LIMITS Prtil sums of the rithmeti series ( := : Arithmeti series with d, S... : Geerl: S d d... S d d... S ( S ( d d d S (( d ( ( d Smll Eerise: Sustitute =, d= i the Geerl formul of S d simplif. Wht do ou oti? Mthemtis Prof. Dr. Philipp E. Zeh 6

.. SEQUENCES, SERIES AND LIMITS Prtil sums of the geometri series ( := : Geometri series S S.q q q q... q S.q S S.(q q S q q... q q q (q (q (q e.g.: q, : S (geerl with (q ( 8 6 Mthemtis Prof. Dr. Philipp E. Zeh 7.. SEQUENCES, SERIES AND LIMITS Emple: Arithmeti Series Suppose tht BMW offers ou to purhse ew model r i mothl istlmets. The first istlmet mouts to 00 d the ever moth the mout of istlmet is iresed. The totl pk period is ers. Wht is the totl prie of the r? The mout of the lst istlmet? The resultig rithmti sequee 00,, 0, 7... We ote tht, = 00, =, d = - 00 =, =. = (for two ers Use geerl S formul to ompute prie of the r d ompositio formul to ompute the vlue of lst istlmet (the lst istlmet is lst term i series. Totl prie of the r: S = (00 + (-((/ = 000 + 6900 = 8900 Amout of lst istlmet: T = 00 + (-( = 07 Mthemtis Prof. Dr. Philipp E. Zeh 8

.. SEQUENCES, SERIES AND LIMITS Eerise: Refer to the dt i lst emple, wht is the mout of moe ou pid i oe er (Yer d Yer? Suppose tht the pk time hs ee iresed to. ers (hit: =0. Wht is the mout whih eh istlmet should e deresed? (hit: ou hve to lulte d ow, give = 00, S = 8900 Wht is the mout of istlmet tht ou mde t the ed of the pk period? (hit: ou re required to lulte the lst term i the sequee. Eerise: A omp iurs ost of.0 for produig uit of some produt. Due to iresed t, for eh suessive uit, the ssoited ost is iresed.0. How m uits e produed i totl of 000? (hit: use the S formul, ou eed to lulte, =., d = Mthemtis Prof. Dr. Philipp E. Zeh 9.. SEQUENCES, SERIES AND LIMITS Emple: Geometri Series The urret prie of some model of Apple MBook is 000. It is epeted to lose its vlue % ever er. Wht would e its vlue i 0th er? Note tht the vlue is lost % i.e. the vlue of omputer t the ed of eh er is 7% of previous er (q = 7% = 0.7. The resultig geometri sequee 000, 00,... = 000, = 00, q = 00/000 = 0.7 The teth term i the sequee is the vlue of omputer i the 0th er 0 =. q 9 = 000(0.7 9 = 0.7 Smll Eerise: Wht is the vlue of omputer t the ed of ers? Mthemtis Prof. Dr. Philipp E. Zeh 0

.. SEQUENCES, SERIES AND LIMITS Emple: Geometri Series Suppose tht ou ivest i rel estte d purhse ld er Hmurg. A reewle eerg usiess firm tkes it o ret. Aordig to terms, the lese e doe for ever si moths, the mout of ret is sujet to 0% irese i eh susequet lese. The first mout of ret is 6000. Clulte the totl mout of retl pmets tht would e reeived for the period of ers. The resultig geometri sequee 6000, 6600, 760... = 6000, = 6600, q = 6600/6000 =., =0 0 6000(. S0 96.. Smll Eerise: Wht is the vlue of retl pmets reeived t the ed of ers? Mthemtis Prof. Dr. Philipp E. Zeh.. SEQUENCES, SERIES AND LIMITS Eerise: Suppose tht produtio plt produes 00 hoolte rs i the first d. The produtio pit e iresed % ever d. Clulte the totl output of the produtio plt for two weeks. (hit: ou hve to use S formul where =00, q=.0 d = Now lso lulte the umer of hoolte rs produed o the 0th d. A super mrket ples order of 800 rs t the strt of the first d of produtio. How m ds will it tke to fulfill the order? Mthemtis Prof. Dr. Philipp E. Zeh

.. POLYNOMIALS Defiitio : A futio with the form i 0... i i0 is lled - thdegreepolomil futio. The rel umers i re lled oeffiiets. Mthemtis Prof. Dr. Philipp E. Zeh.. POLYNOMIALS... st d d degree Polomils Whih differet mthemtil ws e used to desrie geometri ojets? st degree polomils desrie (stright Two- poit - form ( from the slope lies : ( tα d respetivel Geerl form 0 with 0 : iterept : slope Mthemtis Prof. Dr. Philipp E. Zeh

.. POLYNOMIALS... 0th, st d d degree Polomils Emples: The equtio of stright lie (=0, = 0 : 0 for ll : 0 P P P 0 0 0 0 Mthemtis Prof. Dr. Philipp E. Zeh.. POLYNOMIALS... 0th, st d d degree Polomils Emple: Formultio of lier Polomil i Busiess Applitio A firm hs 000 udget to produe two differet produts, mel produt d produt. Eh uit of produt osts 0 d eh uit of produt osts 0. Epress this iformtio i first degree polomil equtio. Let e the umer of uits of produt to e produed. Let e the umer of uits of produt to e produed. Therefore, the ost iurred to produe uits of produt is 0 d vie vers. 0 + 0 = 000 Smll eerise: Suppose tht the firm fford 0 hours of lor. Eh uit of produt requires hours of lor d eh uit of produt requires hours of lor. Epress the iformtio i lier equtio. Tr to sketh the lie. Mthemtis Prof. Dr. Philipp E. Zeh 6

.. POLYNOMIALS The equtio of prol (= Whih differet mthemtil ws e used to desrie geometri oets? d degree polomils desrile prols Two ws of desriig prol: Stdrdformdverteformof theprol(with, verte- oordites: s s 0 d Epdigthe verte form s s S S S S respetivel S S S Verte form of the prol results i the stdrd form. 0 S S Mthemtis Prof. Dr. Philipp E. Zeh 7.. POLYNOMIALS The equtio of prol (= From stdrdform to verte form: ( s s S s S respetivel S S S S S S S S Mthemtis Prof. Dr. Philipp E. Zeh 8

.. POLYNOMIALS The equtio of prol (= Fromverte form to stdrdform: osider,, with verte formprmeters:, epd,we get 6 9 6 stdrdformprmeters: s Emple: s 6, 0 s Emple: From s tdrdform to verte form: 6 0 stdrdformprmeters:, 6, (( 9 (( ( 0 withverteformprmeters:,, s s 0 s Mthemtis Prof. Dr. Philipp E. Zeh 9.. POLYNOMIALS The equtio of prol (= Geerl form: 0 Emples: 0 - - 0 0 0 0 0 0 0 verte t 0 0 0 0 opeig upwrds dowwrds 0, 0. 0 - - 0-0, - 0, - - 0 0 0 0, 0 Mthemtis Prof. Dr. Philipp E. Zeh 0

.. POLYNOMIALS Qudrti equtios (= Zeros of qudrti futio, i.e. itersetio with the -is Or the poits where the vlue of the qudrti futio is etl equl to zero. 0, p q 0, p p q Mthemtis Prof. Dr. Philipp E. Zeh.. POLYNOMIALS d degree polomils (=: It hs to e differetited etwee ses of disrimits: - > 0 => rel zeros Oservtio: Curves ross the -is t etl two poits Mthemtis Prof. Dr. Philipp E. Zeh

.. POLYNOMIALS d degree polomils (=: - = 0 => oe zero Oservtio: Curves touh the -is t etl oe poit. Mthemtis Prof. Dr. Philipp E. Zeh.. POLYNOMIALS d degree polomils (=: - < 0 => o zero Oservtio: Curves do ot ross -is Mthemtis Prof. Dr. Philipp E. Zeh

.. POLYNOMIALS.. Polomils of higher degree rd degree polomil f 0 th degree polomil (Speil iqudrti equtio f (oeffiiets of d rezero Mthemtis Prof. Dr. Philipp E. Zeh.. POLYNOMIALS 0... i0 This is polomil futio. i i A rtiol futio is defied through the rtio of two polomils Z( d N(:.. Polomils of higher degree N-th Degree Polomils Z( N( where Z( is -th degree polomil d N( is m-th degree polomil, for ll IR d N( 0 Mthemtis Prof. Dr. Philipp E. Zeh 6

.. POLYNOMIALS.. Polomils of higher degree Comprig oeffiiets Polomils re idetil if the hve the sme degree d the respetive oeffiiets re idetil. i0 m i i i i i0 i i d m Mthemtis Prof. Dr. Philipp E. Zeh 7.. POLYNOMIALS.. Polomils of higher degree Comprig oeffiiets Emple: 7 ( ( ( 7 ( 7 7 omprig the oeffiiets we get,, 7, 0 Therefore, 0 Mthemtis Prof. Dr. Philipp E. Zeh 8

.. POLYNOMIALS.. Determitio of the zeros of polomil Deomposititio ito lier ftors Theorem: If 0 is the (e.g. guessed zero of -th degree polomil P(, the lier ftor ( 0 e seperted from the polomil: P( = u(( 0. I this se u( is polomil with the degree (. The oeffiiets of the remiig polomil u( e determied through either Horer s method (ot overed here or polomil log divisio. Mthemtis Prof. Dr. Philipp E. Zeh 9.. POLYNOMIALS Polomil log divisio If, i rtiol futio, the degree of the umertor is higher or equl to the degree of the deomitor, polomil log divisio e oduted. Through polomil log divisio the zeros of polomil e determied, if oe zero is lred kow. I this se the origil polomil is divided the respetive lier ftor. Emple: + hs zero t =. Further zeros the e determied through polomil log divisio. I dditito to this slt smptotes e determied, if the degree of the umertor is etl oe higher th the degree of the deomitor (lso see hpter, differetil d itegrl lulus. Mthemtis Prof. Dr. Philipp E. Zeh 60

.. POLYNOMIALS Polomil log divisio Emple: ( + : ( = ( ( 0 Therefore, - is the ftor of ( + Mthemtis Prof. Dr. Philipp E. Zeh 6.. POLYNOMIALS Polomil log divisio Emple: ( + : ( + = ( + ( + 0 Smll Eerise: ( - 9 0 / (+ We kow ( - 9 0 = (+(-0 Verif it with log divisio. Mthemtis Prof. Dr. Philipp E. Zeh 6

.. POLYNOMIALS Properties of polomils A -th degree polomil hs mimum of rel zeros. The dditio, sutrtio, multiplitio d likig of polomil futios (polomils lws results i polomils gi. The divisio, o the other hd, results i rtiol futios. Mthemtis Prof. Dr. Philipp E. Zeh 6. LINEAR ALGEBRA Mthemtis Prof. Dr. Philipp E. Zeh 6

.. SYSTEM OF LINEAR EQUATIONS Lier Equtios: Rell tht we represet stright lie lgerill equtio of the form where, d re rel ostts d re ot oth zero. d re the vriles of the equtio (ofte represetig two differet produts d of firm. Equtio of this form is lled Lier Equtio i.e. the oe i whih vriles hve the power etl equl to d o vriles re multiplied to other vriles e.g. if the term like ppers i equtio, the it will ot e lier equtio. Mthemtis Prof. Dr. Philipp E. Zeh 6.. SYSTEM OF LINEAR EQUATIONS Lier Equtios: Before, we formulted lier equtio for firm tkig deisio o produig umer of uits of two differet produts. Ofte rel life situtios re omple, owds ordir supermrket hs thousds of produts. To desrie suh situtios we eted our ide of lier equtios. For emple, osider situtio if the firm hs to deide o produtio of uits of differet produts, the we write the lier equtio s where (s ovetio...,,,, re ll the vriles of the equtio d,,,, re oeffiiets. These vriles re lso lled s ukows. Mthemtis Prof. Dr. Philipp E. Zeh 66

.. SYSTEM OF LINEAR EQUATIONS Sstem of Lier Equtios: A fiite set of lier equtios is lled sstem of lier equtios or lier sstem. For emple, Sstem of two equtios: Sstem of three equtios: - z z - z 9 Mthemtis Prof. Dr. Philipp E. Zeh 67.. SYSTEM OF LINEAR EQUATIONS Sstem of Lier Equtios: Sstem of lier equtios (geerlizig the otio: A ritrr sstem of m equtios with vriles e give :...... m m m... m m I order to ler the Algorithm (step step proedure used to solve lier sstems, we will go through some prelimir oepts o Mtries. Mthemtis Prof. Dr. Philipp E. Zeh 68

.. SYSTEM OF LINEAR INEQUALITIES Defiitio (p. Opitz, p. : A sstem of iequlities of the form + + +... + + + +... + m + m + m +... + m m... Is lled lier sstem of iequlities with ukows (vriles,..., d m equtios. The vlues ij d i (i =,..., m, j =,..., re give d the re lled the oeffiiets of the sstem of iequlities. Sought fter re the vlues for the vriles,...,, so tht ll iequlities re fulfilled simulteousl. Mthemtis Prof. Dr. Philipp E. Zeh 69.. SYSTEM OF LINEAR INEQUALITIES Ever sstem of iequlities with reltioships of iequlit (, d equlit e rerrged to the ove form. Possile rerrgemets re: Multiplitio - Deompose oe equtio ito two iequlities Defiitio: The set of ll ssigmets of vlues (vetors IR, whih fulfill erti sstem of iequlities, is lled solutio set or dmissile rge of the sstem of iequlities. Mthemtis Prof. Dr. Philipp E. Zeh 70

.. SYSTEM OF LINEAR INEQUALITIES Emple: A mufturer produes two tpes of mouti ikes, tpe Sport (S d tpe Etr (E. Durig the produtio ever ike goes through two differet workshops. I shop : 0 workig hours re ville eh moth; i shop : 80 hours re ville. To produe tpe S ike si hours re eeded i shop A d three hours re eeded i shop B. For tpe E ike four d te hours re eeded respetivel. Wht is the respetive sstem of iequlities? Show the dmissile rge grphill! Mthemtis Prof. Dr. Philipp E. Zeh 7.. SYSTEM OF LINEAR INEQUALITIES Deisio vriles: A deisio hs to e mde out the mout whih should e produed of oth tpes of ikes. S = Mothl mout produed of tpe S ikes. E = Mothl mout produed of tpe E ikes. Coditio for produer : required time <= ville time 6S + E <= 0 Coditio for produer : required time <= ville time S + 0E <= 80 No-egtivit: S >= 0, E >= 0 Mthemtis Prof. Dr. Philipp E. Zeh 7

.. SYSTEM OF LINEAR INEQUALITIES E produer fesile regio 0 produer 0 0 60 S Mthemtis Prof. Dr. Philipp E. Zeh 7.. SYSTEM OF LINEAR INEQUALITIES Three ses hve to e distiguished: Norml se: The spe of dmissile solutios Z is ouded d ot empt. The it is ove polhedro (s i the ove emple. There is o solutio for the sstem of iequlities. The spe of dmissile solutios Z is empt. The spe of dmissile solutios is uouded. Mthemtis Prof. Dr. Philipp E. Zeh 7

.. SYSTEM OF LINEAR INEQUALITIES Coveit of the dmissile rge: ot ove ove Coveit mes tht ever oetig lie etwee two poit of the dmissile rge ompletel lies iside the rge. Mthemtis Prof. Dr. Philipp E. Zeh 7. Differetil Clulus Mthemtis Prof. Dr. Philipp E. Zeh 76

.. BASICS The (ostt slope of stright lie is equl to the derivtive of the orrespodig futio f( = = +. The situtio for o-lier futios (urves is more omplited, sie the slope is ot equl t ever poit, ut hges depedig o. Slope of urve: Slope t poit P Therefore, to determie the slope t erti poit, ou hve to look t the tget of the urve i this poit. Tget: A stright lie, whih touhes urve t erti poit, ut whih does ot iterset the urve. Mthemtis Prof. Dr. Philipp E. Zeh 77.. BASICS EXAMPLE : TANGENT TO A CURVE The slope of tget t poit gives the hge i with respet to (mrgil hge i Tget t poit * Mthemtis Prof. Dr. Philipp E. Zeh 78

Mthemtis Prof. Dr. Philipp E. Zeh 79 Emples: Differet tgets to urve = 8 slope t * = : d/d = slope t * = : d/d =.. BASICS Mthemtis Prof. Dr. Philipp E. Zeh 80 Power Futio Emples: 9 ( ( 0 ( ( 8 9 9 9 0 0 0 7 7 7 7 7 7.. BASICS

.. BASICS Epoetil Futio e ( e l e l e e l l l Applitio: Clulte f : f = Logrithmi Futio d Trigoometri Futio l 0 log l l l ' l (l ' si os os si Mthemtis Prof. Dr. Philipp E. Zeh 8.. DERIVATIVE RULES Ftor Rule: f ( f ( Emples: 0 ( 0 0 0 ost Sum Rule: f ( g( f ( g( Emples: 7 e l ' ' e Mthemtis Prof. Dr. Philipp E. Zeh 8

Mthemtis Prof. Dr. Philipp E. Zeh 8 Produt Rule: h g f g f g f ( ( ' ' ' ' ' ' ' h g f h g f h g f g f g f Emple: (l l ( (l ( ( ( e e e e e.. DERIVATIVE RULES Mthemtis Prof. Dr. Philipp E. Zeh 8 d Quotiet Rule: ( ( g f ' ' ' g g f g f Emple: ( ( ( ( ( (.. DERIVATIVE RULES

Mthemtis Prof. Dr. Philipp E. Zeh 8 e Chi Rule: ( ( z g f '( ( '( '( ( ' g g f g f d dz dz df Emple: ( ( (0 ( e ' '( ( e e h g f.. DERIVATIVE RULES Mthemtis Prof. Dr. Philipp E. Zeh 86 Emple: 0... 7 7 6 7 7 (7 (6 Derivtives of higher order re eessr for the solutio of optimiztio prolems (urve skethig.. APPLICATIONS & EXERCISES CURVE SKETCHING

.. APPLICATIONS & EXERCISES CURVE SKETCHING Curve skethig gives iformtio out the hrteristis d the ehviour of the prtiulr futio, of profit, reveue or ost futio. Domi (-vlue d (if eessr odomi (-vlues Smmetr hrteristis Smmetr out the -is f(=f(- e.g. = Poit smmetr out the origi f(=f(- e.g. = Itersetio with the es Itersetio with the -is: =0; f(0 orrespodig -vlue Itersetio with the -is: Zeros. f(=0; solve for d Gps d poles (vertil smptotes e.g. i the se of rtiol futios (quotiet of two polomils Z(/N( gps i the defiitio eist i the zeros of N(. Mthemtis Prof. Dr. Philipp E. Zeh 87.. APPLICATIONS & EXERCISES CURVE SKETCHING e Asmptotil ehviour d smptotes: Asmptotil ehviour (horizotl smptotes: Behviour of the futio if d if - : Does the futio ted to, - or to ostt vlue? Vertil smptotes: If the gp i the defiitio 0 is pole, the the stright lie = 0 is vertil smptote of the grph of f. Slt smptotes: If futio pprohes stright lie more d more with iresig, the this lie is lled slt smptote. Rtiol futios hve smptotes, if the degree of the umertor is mimum oe higher th the degree of the deomitor. The the lier equtio of the smptote e foud through polomil log divisio. The term i frot of the remider shows the lier equtio of the smptote. Mthemtis Prof. Dr. Philipp E. Zeh 88

.. APPLICATIONS & EXERCISES CURVE SKETCHING Coetio etwee Etrem d Derivtive: Oservtio: If the futio shows lol mimum, the slope of the urve must e positive to the left d egtive to the right of the mimum. I.e., the futio vlues irese efore the mimum d derese fterwrds. Colusio: At the mimum the slope (= vlue of the derivtive is etl zero! slope= 0 slope positive slope egtive Lol mimum: There is eviromet i whih o poit hs higher futio vlue. Mthemtis Prof. Dr. Philipp E. Zeh 89.. APPLICATIONS & EXERCISES CURVE SKETCHING f Slope d etrem: mim, miim d sddlepoits etrem (horizotl tget f ( E 0 lol miimum E f ( E 0 f ( E 0 lol mimum E f ( E 0 f ( E 0 sddle poit f ( E 0 The seod derivtive shows hge i the slope. Seod derivtive positive: slope ireses => lol miimum Seod derivtive egtive: slope dereses => lol mimum Seod derivtive is equl to zero: either mimum or miimum, ut sddle poit. Mthemtis Prof. Dr. Philipp E. Zeh 90

.. APPLICATIONS & EXERCISES CURVE SKETCHING Neessr d suffiiet oditios for lol etrem Neessr oditio for lol mimum or miimum: f ( 0 = 0 I.e., if there is mimum or miimum t 0, the f ( 0 = 0 Suffiiet oditio for lol mimum: f ( 0 = 0 d f ( 0 < 0 I.e., if f ( 0 = 0 d f ( < 0, there is lol mimum t 0 Suffiiet oditio for lol miimum: f ( 0 = 0 d f ( 0 > 0 I.e., if f ( 0 = 0 d f ( > 0, there is lol miimum t 0 Mthemtis Prof. Dr. Philipp E. Zeh 9.. APPLICATIONS & EXERCISES CURVE SKETCHING g Curvture d ifletio poits f ( W 0 f ( W 0 Ifletio Poit f ( 0 f ( 0 At the ifletio poit urvture to the right hges to urvture to the left f ( 0 f ( 0 At the ifletio poit urvture to the left hges to urvture to the right Mthemtis Prof. Dr. Philipp E. Zeh 9

.. APPLICATIONS & EXERCISES CURVE SKETCHING Emple: f( = (- ( 6 Domi IR No smmetr X to +/-, f( to - f ( ( f ( f ( ( 6 8 ( 6 f ( 8 ( f ( 8. Zeros: f ( 0 ( ( 6 0 00 6 (oudr poit 0 ( 8 8 00 Mthemtis Prof. Dr. Philipp E. Zeh 9.. APPLICATIONS & EXERCISES CURVE SKETCHING Cotiutio of the Emple:. Etrem: f( 0 ( 6 0 0 6 M ormi? f ( f (0 0 NEITHER f ( f (6 MAX t (6,0 (q.v. ZEROS!. Ifletio poits: f ( 0 ( 0 0 f(0 f( 76 Mthemtis Prof. Dr. Philipp E. Zeh 9

.. APPLICATIONS & EXERCISES CURVE SKETCHING Cotiutio of the Emple: Nture of the hge f ( f ( f(0 0 f (0 0 f (0 0 f (0 8 0 r/l f ( 8 0 l/r SP t (0,-, r/l WP t (,-76, l/r Mthemtis Prof. Dr. Philipp E. Zeh 9.. APPLICATIONS & EXERCISES CURVE SKETCHING Cotiutio of the Emple. Sketh: Tget t P(-,- => t (=8-6 Tget t IP=> t (=8-688 - 6 MAX P -6 r/l SP - - IP l/r f( Mthemtis Prof. Dr. Philipp E. Zeh 96

.. APPLICATIONS & EXERCISES CURVE SKETCHING Cotiutio of the Emple:. Sketh of the derivtive - 6 f( Mthemtis Prof. Dr. Philipp E. Zeh 97.. APPLICATIONS & EXERCISES CURVE SKETCHING Cotiutio of the Emple. Sketh of the seode derivtive - 6 f( Mthemtis Prof. Dr. Philipp E. Zeh 98

.. APPLICATIONS & EXERCISES CURVE SKETCHING Cotiutio of the Emple. Tget t IP: Slope t IP: f( 8 t( m 76 8 688 t ( 8 688 Mthemtis Prof. Dr. Philipp E. Zeh 99.. APPLICATIONS & EXERCISES CURVE SKETCHING 6. Prllel lie to the ifletio tget: f( 8 8 8 0 ( 8: ( ( Boudr poit P (, f( Tget t P(, m 8 t( m 8( 6 t( 8 6 Mthemtis Prof. Dr. Philipp E. Zeh 00

. Itegrl Clulus Mthemtis Prof. Dr. Philipp E. Zeh 0.. BASICS. Theorem of differetil d itegrl lulus If f( is itegrted, ou oti the primitive futio F(. If F( is differetited, ou oti f(. Itegrtio is the reversl of differetitio.. Theorem of differetil d itegrl lulus If F( is the primitive futio of f(, the the defiite itegrl is F( F( F( f ( d Defiitio: the geerl form F(+ ( = ostt of itegrtio of primitive futio f( is lled idefiite itegrl of f(. You write: f d F( ( Without limits, Mthemtis Prof. Dr. Philipp E. Zeh 0

.. BASICS ADVISE FOR INTEGRATING: If futio is otiuous, it is lso itegrle. If futio is pieewise otiuous, primitive futio e determied for ever otiuous itervl. The defiite itegrls re lulted oe--oe for the prtil itervls d the dded up to the omplete itegrl. Aout the lultio of re: Durig the lultio of re futios ot e itegrted eod zeros. If there re zeros, it hs to e itegrted i setios. I order to determie the whole re etwee the urve d the -is, it hs to e itegrted over the solute vlue of the futio i the egtive setios. The ll vlues hve to e dded up. Mthemtis Prof. Dr. Philipp E. Zeh 0.. BASICS Power Futio: f ( d - Testig Differetitio: ( ( Mthemtis Prof. Dr. Philipp E. Zeh 0

Mthemtis Prof. Dr. Philipp E. Zeh 0 d d d d Emples: 0 Testig differetitio:.. BASICS Mthemtis Prof. Dr. Philipp E. Zeh 06 Epoetil futios: 0 für ( l( l ( 0 für l f( l( ( l( f( d f e d e e d.. BASICS

.. RULES FOR INTEGRATION Sum Rule: f( g( ( f ( g( d f ( d g( d Rule: A sum of futios e itegrted oe oe! Emple: si ( si d d si d os Mthemtis Prof. Dr. Philipp E. Zeh 07.. RULES FOR INTEGRATION Ftor Rule: f ( d f ( d Rule: A ostt ftor e moved i frot of the itegrl durig itegrtio! Emple: ( d d Mthemtis Prof. Dr. Philipp E. Zeh 08

Mthemtis Prof. Dr. Philipp E. Zeh 09 ( d 7 9 ( d Emple : Emple : 6 ( d e Emple : Prtil itegrtio d itegrtio sustitutio re ot overed here... APPLICATIONS & EXERCISES