Download - Explorations for the First Week of Calculus · Students investigate the motion of a door swinging open and closed, first by sketching a reasonable graph of number of degrees, d(t),

ExplorationsfortheFirstWeekofCalculusByPaulA.Foerster,[email protected]

HerearefourExplorations(includingsolutions)thatIwrotetogivemystudentsatasteofthe

fourconceptsofcalculus(limits,derivatives,integrals,integrals)duringthefirstweekofthe

course.Theconceptsaredevelopedgraphically,numerically,andverbally,usingaccurate

graphsonpaper,verifiedbygraphingcalculator.YouarewelcometousetheseExplorationsin

variousways(includingcooperativegroupsorflippedclassroom)fordirectteachingofstudents

inyourschool.Pleasedonotusetheminanywaythatwouldinfringeonthecopyrights.

• Exploration1‐1a:InstantaneousRateofChangeofaFunction(firstdayofcalculus)

Studentsinvestigatethemotionofadoorswingingopenandclosed,firstbysketchinga

reasonablegraphofnumberofdegrees,d(t),thedoorisopen,versusnumberofseconds,t,

sinceitwaspushed.Thentheyinvestigatethefunctiond(t)=200t(2–t)thathasagraph

similartotheonetheyhavedrawn.Theyestimatethe“instantaneousrate”att=1by

calculatingaverageratesoversmallerandsmallertimeintervalsaboutt=1.Theylearnthat

aderivativeisthelimitoftheaverageratesofchangeasthetimeintervalapproacheszero.

TheycangohomeonDay1andsay,“HeyMomandDad!Ididcalculustoday!”

• Exploration1‐2a:GraphsofFunctions(Day2)

Studentsgetabriefreviewofgraphsofdifferentkindsoffunctionsfromprecalculus.

• Exploration1‐3a:IntroductiontoDefiniteIntegrals(Day3)

Studentsinvestigatethemotionofacarasitslowsdownafterpassingatruckthen

continuesataconstantvelocity.Theyobservethatwhenthevelocityisconstant,(distance)

=(velocity)(time),andthusthedistancetraveledisequaltotheareaofaregionunderthe

velocity‐timegraph.Fromagivenaccuratelyplottedgraphonpaper(whoseequationisnot

given),theyestimatethedistancethecargoeswhileslowingdownbycountingsquares

underthecurvedportionofthegraph.TheylearnonDay3thatadefinite integralgivesa

wayofmultiplyingthey‐variablebythex‐variableifyvarieswithx.(Indefiniteintegralsand

thefundamentaltheoremcomelater,aftertheylearnandapplyderivativeformulas.)

• Exploration1‐4a:DefiniteIntegralsbytheTrapezoidalRule(Days4and5)

Studentsinvestigatethemotionofahypotheticalspaceshipasitspeedsupandslows

down,firstbycountingsquaresunderthegivenaccurately‐drawnvelocity‐timegraphasin

Exploration1‐3a.Thentheyaregiventheequationofthegraph,dividetheregionunderthe

graphintofourtrapezoids,andcalculatebyhandthesumoftheareas.Nexttheydownload

aprogramtofindthesumoftheareasforntrapezoids.Theyobservenumericallythatthe

definite integralisequaltothelimitofthesumoftheareasasnapproachesinfinity.

TheExplorationsaccompanymytextCalculus:ConceptsandApplications,originallypublished

byKeyCurriculumPressandnowwithKendallHunt.Fora30‐dayfreetrialofthetext,including

the150+Explorations,gotowww.flourishkh.com,clickon“PreviewNow,”under

“Mathematics,”clickon“Calculus,”thenclickon“Requestatrial.”

PaulA.Foerster,TeacherEmeritusofMathematics

AlamoHeightsHighSchool,SanAntonio