This processof findinghow functionvaluesbehaveas we systematicallyget closerand closerto partic-
ular -valuesis the processof finding limits. In the next lessonwe will formallydefinethis processand
developsomeefficientwaysfor computinglimitsof functions.
Applicationsof DifferentialCalculus
Maximizingand MinimizingFunctions
RecallfromLesson1.3 our exampleof modelingthe numberof FoodStamprecipients.The modelwas
foundto be with graphas follows:(Useviewingwindowrangesof on
and on )
We note that the functionappearsto attaina maximumvalueaboutan -valuesomewherearound
Usingthe processfrom the previousexample,whatcan we say aboutthe tangentline to the graphfor that
valuethat yieldsthe maximum value(the pointat the top of the parabola)? (Answer:the tangent
line will be horizontal,thushavinga slopeof 0.)
Hencewe can use calculusto modelsituationswherewe wish to maximizeor minimizea particularfunction.
This processwill be particularlyimportantfor lookingat situationsfrom businessand industrywherepolynomial
functionsprovideaccuratemodels.
Velocityof a FallingObject
We can use differentialcalculusto investigatethe velocityof a fallingobject.Galileofoundthat the distance
traveledby a fallingobjectwas proportionalto the squareof the time it has beenfalling: