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1、ChapterExponentialandLogarithmicFunctionsTheMh~derivativetestdevelopedinChap.9equipsusforthetaskoflocatingtheextremevaluesofanyobjectivefunction,uslongasitinvolvesonlyonechoicevariable,possessesderivativestothedesiredorder,andsoonel'orlateryieldsanonzeroderivativevalueatthecritic
2、alvaluexo.lntheexamplescitedinChap.9,however,wemadeuseonlyofpolyno~mialandrationalfunctions,forwhichwekn()whowtoobtainthenecessaryderivatives.Supposethatourobjectivefunctionhappenedtobeunexponentialone,suchasy=gt-,[tThenwearestillhelplessinapplyingthederivativecriterion,becausewe
3、haveyettolearnhowtodifferentiatesuchafunction.Thisiswhatweshalldointhepresentchapter,Exponentialfunctions,aswellasthecloselyrelatedlogarithmicfunctionr.;,haveimpor~tantapplicationsineconomics,especiallyinconnectionwithgrowthproblems.andineco•nomicdynamicsingeneral.Theparticularap
4、plicationrelevanttothepresentpartofthebook,however,involvesaclassofoptimizationproblemsinwhichthechoicevariableistime.Forexample,acertainwinedealermayhaveastockofwine,themarketvalueofwhichisknowntoincreasewithtimeinsomeprescribedfashion.Theproblemistodeter•minethebestlimetosdltha
5、tstockonthebasisofthewinewvaluefunction,aftertakingintoaccounttheinterestcostinvolvedinhavingthemoneycapitaltiedupinthatstock.Expo•nentialfunctionsmayenterintosuchaproblemintwoways,First,thevalueofthewinemayincreasewithtimeaccordingtnsomeexponential/awofgrowth.Inthatevent,\-ewou
6、ldhaveanexponentialwine-valuefunction.Second,whenweconsidertheinterestcost,thepresenceofinterestcompoundingwillsurelyintroduceanexponentialfunctionintothepicture.Thuswemuststudythenatureofexponentialfunctionsbeforewecandiscussthistypeofoptimizationproblem,Sinceourprimarypurposeis
7、todealwithtimeasachoicevariable,letusnowswitchtothesymbolt~inlieuofx~toindicatethcindependentvariableinthesubsequentdiscus•sion,(However,thissamesymboltcanvcrywellrepre~entvariablesotherthantimealso.)2SS256PMtFourOpfimi:.atioltP!!h/ems10.1TheNatureofExponentialFunctionsAsintrodu
8、cedinconnectionwithpolynomialfunctions,t