LawrenceWoodmereAcademy
APCalculusAB
DearAPCalculusABStudent,
WelcometothefunandexcitingworldofAPCalculusAB.Intheupcomingschoolyear,wewillbeusingtheconceptsthatyoupreviouslylearnedinintegratedalgebra,geometry,algebraII,trigonometryandpre-calculustoexpandyourknowledgeintotheworldCalculus.TobeabletomoveforwardinABCalculus,youmusthaveastrongfoundationintrigonometry,conceptsinvolvingfunctions,andbeabletocreatemodelsfromwordproblems.Thissummerassignmentisdesignedtoallowyoutocontinuetopracticetheseskillsandconceptsthroughoutthetimethatschoolisnotinsession.
TheAPCalculusABSummerAssignmentpacketwillnotrequirealotoftime,butitislengthyenoughthatyouwillwanttomanageyourtimeappropriately.Thewholeassignmentshouldnotbecompletedattheendofthisschoolyear,butshouldbeworkedonallsummertokeepthematerialfreshinyourmind.AsAPCalculusABstudents,youwillneedtobeabletomanageyourtimeappropriately.
Thissummerassignmentiscomposedoftwosectionsthatreviewtheoldmaterialfromyouryearsinhighschool.Thefirstsectionisareviewoftrigonometry.Makesurethatyoumakeyourselffamiliarwithalltheexactvaluesfortrigonometricfunctionsontheinterval[0,2𝜋].ThereisaNOcalculatorsectionontheAPandyoumusthavethesevaluesmemorized.Thesecondsectionfocusesonareviewoffunctionsandmodeling.Youareexpectedtoanswerallquestionsonaseparatesheetofpaperandhandintheassignmentonthefirstdayofschool.Allworkmustbeshownforeachofthequestionsandyoumustprovideexplanationsforallmultiplechoicequestions.It’snotsufficientenoughtogettherightanswer,butyoumustbeabletoexplainyouransweraswell.Theassignmentwillbegradedforcompletionandeffort.
Youshouldalsogettherequiredsuppliesforthecoursewhichincludesgraphpaper,notebook,pencil,andaTI-83orTI-84.YoumayalsowanttogotoabookstorethissummerandpickupanAPpreparationguidefortheAPexam(IrecommendBaron’s).
Ifyouhaveanyquestions,donothesitatetoe-mailmeoverthesummeratdearley@lawrencewoodmere.org
Again,welcometoAPCalculusAB!
GoodluckandIlookforwardtoseeingyouinSeptember.
Sincerely,
Mrs.DanielleEarley
TRIGONOMETRY:1.Usethediagramontherighttofindtheexactvaluesofthefollowing:
a. tanA
b.cosB2.Whichofthefollowingisequaltocscθ?
a. !!"#$
b. !
!"#$
c. !!"#$
d. !
!"#$
3.Findtheexactvalueofsec 300°.4.Findthereferenceangleforananglemeasuring 145°.5.Whichofthefollowinggraphsrepresentstheequation𝑦 = −1+ sin 2𝑥 overaone-periodinterval?a.
b.
c.
d.
6.Change125°toradianmeasure.7.Change!"!
!" radianstodegreemeasure.
8.Findtheexactvalueofeachofthefollowing:a.sin !!
!
b.tan !! c.cos !
!
d.sec𝜋
e.csc !!! f.cot !!
!
g.sin!! ! !! h.cos!! !
!
i.tan!! 3
**Youmustknowtheexactvaluesoftrigonometricfunctionsbyheart.ThereisaportionoftheAPCalculusexamthatdoesNOTallowcalculatorsandthesevaluesarenecessaryinthatsection.Youshouldhavethemmemorizedbythebeginningoftheschoolyear.FUNCTIONS:1.Giventhatf(x)=2x2+x−5,findf(−3).2.For𝑓(𝑥) = 𝑥 + 5and𝑔 𝑥 = 3𝑥 + 1, findthedomainof!
!.
a. −∞,∞ b. −∞,− !
!∪ − !
!,∞
c. −∞, 5 ∪ 5,∞ d. −∞,− !
!∪ − !
!, 5 ∪ 5,∞
3.Findthedomainfor𝑓 𝑥 = 2𝑥 + 5a. −∞,− !
!∪ − !
!,∞
b. −∞, !!∪ !
!,∞
c. −∞,− !!
d. − !!,∞
4.For𝑓 𝑥 = 2𝑥 − 5and𝑔 𝑥 = 𝑥! − 6find𝑓(𝑔 𝑥 )a.4𝑥! − 20𝑥 + 19b.2𝑥! − 11c. 2𝑥! − 17d.2𝑥 − 115.Writetheequationoftheverticalasymptoteof𝑓 𝑥 = !!!
!!!.
6.Thegraphoftheequation!!!!!!!"!!!
isalinewithaholeinit.Atwhatpointdoestheholeoccur?
7.Identifythepossibleformulaforthegraphshownontheright.a.𝑦 = (𝑡 + 1)(𝑡 − 4)(𝑡 − 3)b.𝑦 = (𝑡 − 1)(𝑡 − 4)(𝑡 + 3)c. 𝑦 = (𝑡 + 1)(𝑡 + 4)(𝑡 + 3)d.𝑦 = −(𝑡 + 1)(𝑡 + 4)(𝑡 − 3)8.Whichformulabestmatchesthegraphshownontheright.a.𝑦 = 𝑥! − 3 + 2b.𝑦 = (𝑥 − 2)! + 3c. 𝑦 = 𝑥! + 3 − 2d.𝑦 = (𝑥 − 3)! + 29.Forthefunction𝑔 𝑥 = −8+ 4𝑥 + 3𝑥! − 𝑥!,whatistheleadingcoefficient?10.Findtherootsofthefollowingequation.Givethevaluesinexactform.
𝑥! + 24𝑥 = 25𝑥!11.Findthevertexforthefollowingequation.
𝑥! − 8𝑥 − 𝑦 + 18 = 012.Findaformulafortheinverseof
a.𝑓 𝑥 = 4𝑥 + 3.b. 𝑓 𝑥 = !
!!!!
13.Suppose$8000isinvestedata4%interestrate,compoundedmonthly.Howmuchwilltheinvestmentbeworthafter9years?14.Evaluatethelogarithm:log!
!!
15.Simplifyusingtherulesoflogarithms:log!25+ log! 3
a.log! 28b.log!
!"!
c. log! 75d.log! 28
16.Expandthefollowingassumsand/ordifferencesofsimplerlogarithmicexpressions.
𝑙𝑛3𝑥 𝑥2𝑥 + 1 !
a.ln 3𝑥 + !!ln 𝑥 − 2 ln(2𝑥 + 1)
b.2 ln 2𝑥 + 1 − ln 3𝑥 + !!ln 𝑥
c.3 ln 𝑥 + ln 𝑥 − 2 ln(2𝑥 + 1)d.ln 3𝑥 + 2 ln 𝑥 − !
!ln(2𝑥 + 1)
17.Converttoanexponentialequation:logx=15a.𝑒!" = 𝑥b.10!" = 𝑥c.15!" = 𝑥d.𝑥!" = 1018.Thepopulationofbacterialculturedoubledin8hours.Whatwastheexponentialgrowthrate?a.3.8%b.4.2%c.5.5%d.8.7%19.Findtheindicatedtermofthegeometricsequence100,80,64,...,a6a.!",!"#
!"#
b.!,!"#!"
c.!,!"#!"#
d.!.!"#!"
20.Findthesumofthefirst36termsinthearithmeticseries:−0.2+0.3+0.8+...a.318.6b.332.2c.307.8d.31421.Findthexandyinterceptsforeachgraph.a.𝑦 = 2𝑥 − 5
b.𝑦 = 𝑥! + 𝑥 − 2
c.𝑦 = 𝑥 16− 𝑥!
d.𝑦! = 𝑥! − 4𝑥
22.Findtheintersectionpointsofthegraphsforthegivenequationsa.𝑥 + 𝑦 = 8 4𝑥 − 𝑦 = 7
b.𝑥! + 𝑦 = 6 𝑥 + 𝑦 = 4
c.𝑥 = 3− 𝑦! 𝑦 = 𝑥 − 1
d.𝑦! = 1− 𝑥! 𝑦! = 𝑥! − 3𝑥 + 2
23.Ifaandharerealnumbers,findandsimplify𝑓(𝑎), 𝑓(𝑎 + ℎ), ! !!! !!(!)
!when:
a. 𝑓 𝑥 = 𝑥! − 𝑥 + 3
b. 𝑓 𝑥 = !!
24.Dividebyusinglongdivision.
a. 2(20 13 2) (4 1)x x x− + ÷ − b. 2( 2 3) ( 5)x x x− + ÷ +
c. 3 2( 2 2) ( 2)x x x x+ − − ÷ + d. 2(6 7 5) (3 5)x x x− − ÷ −
25.Dividebyusingsyntheticdivision.
a. 2(7 23 6) ( 3)x x x− + ÷ − b. 4( 5 10) ( 3)x x x− + ÷ +
c. 2 1(2 13 8) ( )2
x x x+ − ÷ − d. 4 3 2( 6 6 ) ( 5)x x x x+ + ÷ +
26.27.Solvethefollowingequationsfory'
a. 2𝑥 + 2𝑥𝑦’ = 2𝑦 + 3𝑦!𝑦′
b. !!!!!
= !!!
28.Findthezeroesofthefunction(algebraically):𝑓 𝑥 = −𝑥!𝑒!! + 2𝑥𝑒!!
29.TriangleABChasverticesA(0,0),B(4,8),C(10,0)
a.FindthecoordinatesofM,themidpointofsegment𝐴𝐵b.FindtheequationofthelinethatcontainsMandisparalleltosegment𝐵𝐶c.FindanequationofthelinethroughpointsCandM.Isthislineperpendiculartobisector𝐵𝐶
30.Usepoints(-2,4)and(6,2)
a. Findtheslopeofthelinecontainingthesepoints.b. Findthelengthofthesegmentthatconnectsthesetwopointsc. Findthemidpointofthelinesegmentthatconnectsthepoints
a.
b.
31.Findthelinethatpassesthrough(-2,4)andthepointofintersectionofthelinesx+3y=1and2x–y=5.
32.Describehoweachofthefollowinggraphscomparetoitsbasegraph
a. 𝑦 = 2 𝑥 + 2 ! − 4b. 𝑦 = −2 𝑥 − 3 + 6c. 𝑦 = !
!!!+ 4
d. 𝑦 = !!3− 𝑥 + 1
33.Findthevertexandtheaxisofsymmetryfortheparabola𝑦 = 2𝑥! + 8𝑥 + 5
a.Rewritethegraphinvertexform𝑦 = 𝑧 𝑥 − ℎ ! + 𝑘
b.Supposethisgraphisshifted3unitsleftand2unitsuprewritethenewgraphinbothvertexformandstandardform.
34.YouneedaLearJetforoneday.KnowingthatSwissairrentsaLearjetwithapilotfor$3000adayand$1.25permile,whileAirFrancerentsaLearjetwithapilotfor$2500adayand$2.10permile,findthefollowing
a.Foreachcompany,writeaformulagivingthecostasafunctionofthedistancetraveledb.AtwhatMileageisthepricegoingtobethesameforbothSwissairandAirFrancec.Whatcanyouconcludefromthis?
35.In1984,theFizzyColaCo.sold23milliongallonsofsoda.By2003,thecompanywasselling127milliongallonsofsoda.Whatistheaveragerateofchangeinnumberofgallonsofsodaperyear?
36.Thesidesofarectangleaxand3-2x.Expresstherectangle’sareaasafunctionofx.Expresstherectangle’sperimeterasafunctionofx.Whycanxnotequal2?
37.Theheightandthediameterofacylinderareequal.Expressthevolumeofthecylinderasafunctionofitsradius.
38.Eachlegofanisoscelestriangleistwiceaslongasitsbase.Expresstheperimeterofthetriangleintermsofthelengthofthebase(b).
