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

dearley@lawrencewoodmere.org

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).