CONGRATULATIONS!!YoumadeittoBCCalculus!

Iamlookingforwardtocontinuingtoworkwithyouduringthe2017-2018schoolyear.

InordertocompletethecurriculumbeforetheAPExaminMay,itisnecessarytodosomepreparatoryworkthissummer.Thispacketwillhelpyoutofocusonthemathematicalskillsandcontentyouwillneedinordertobesuccessful.TheseproblemsdealwithcertainskillsandcontentthatyouhavepreviouslystudiedinAlgebra2/TrigandHonorsPrecalculus.Youcanuseanynotesyouhavetohelpyousolvethereviewproblems.Therearealsomanywebsitesontheinternetthatcanbehelpfulincompletingtheseproblems,suchaswww.wolframalpha.com,www.khanacademy.org,www.purplemath.com,www.korpisworld.com

PacketsWILLBECOLLECTEDonWednesdaySeptember6th.Theassignmentwillcountasatestgradeinthefirstquarter.

Inordertoreceivefullcredit,youmustcompletetheworkonSEPARATEpaperandyoumustshowALLworktojustifyyouranswersevenifitisamultiplechoicequestion.Thereisablankanswersheetavailableforprintingonmywebsite,www.calcnerd.com,whichcanbeusedtohelporganizeyourwork.Acalculatormayonlybeusedontheindicatedsections.Atthislevel,doinghomeworkismorethanjustgettingtheproblemsdoneitispartofthelearningexperience.Throughouttheyearwewillbeworkingtogetherasaclasstohelpeveryoneachievetheirgoals.SothissummerIencourageyoutoworktogetherwithotherclassmateswhowillbetakingthecoursewithyou.WhatcouldbebetterthandoingCalculusatthebeach?Ifyouhaveanytroublewiththeassignmentorjusthavequestionsorconcerns,youcanreachmebyemailatcrosado@wfsd.k12.ny.us.Iwillrespondassoonaspossible.Istronglyrecommendedthatyoudoafewproblemseachdaythroughoutthesummer.Donotleavetheentireassignmentforthenightbeforeschoolstarts.Havefunandenjoyyoursummer.IwillseeyouinSeptember.

Sincerely,

ChristineRosadoMathematicsDepartmentChairpersonAPCalculus/PrecalculusTeacherNationalBoardCertifiedTeacher(NBCT)

Directions:Pleasereadeachquestioncarefullyandrecordyouranswersonaseparatesheetofpaper.YoumustshowALLWORKinordertoreceivefullcreditandanswersshouldbeCLEARLYmarked.IfthequestiondoesnotrequireworkthengiveanEXPLANATIONforyouranswer.

Part1:Let’sTalkPre-Calculus:1)Obtainthepartialfractiondecompositionforeachofthefollowing:a) !

"#$"%&b) !"$''

"#%"%(c) '%!"

!"#%)"$&

2)Evaluateeachofthefollowinglogarithms:a)𝑙𝑛𝑒)b)ln '

0c)𝑒!12&d)𝑙𝑛 𝑒

3)Rewrite'

&𝑙𝑛 𝑥 − 3 + 𝑙𝑛 𝑥 + 2 − 6𝑙𝑛𝑥asasinglelogarithmicexpression.

4)Findtheexactvalueforeachofthefollowing. a)𝑠𝑖𝑛 ;<

(b)𝑐𝑠𝑐 <

!c)𝑐𝑜𝑠 &<

!

d)𝑠𝑒𝑐 − &<

!e)𝑡𝑎𝑛 <

&f)𝑐𝑜𝑡 − !<

A

5)Let𝑓 𝑥 = 2𝑥 + 1𝑎𝑛𝑑𝑔 𝑥 = 2𝑥& − 1.Findeachofthefollowing:a)𝑓(𝑡 + 1)b)𝑓(𝑔(−2))c)𝑔 𝑓 𝑚 + 2 d) 𝑓(𝑥) & − 2𝑔(𝑥)6)Solvefor𝑡:(1.045)O = 27)Thepopulationofabacterialculturedoubledineighthours.Whatwastheexponentialgrowthrate?

8)Thenumberofbacteriainaculturewas1000onMondayandthenumberhasbeendoublingeverydaysincethen.a.Writeafunctiontomodelthebacterialpopulation.b.Predictthenumberofbacteriainthecultureinoneweek.Includeunits.c.Howlong,tothenearesttenthofaday,willittakethebacterialculturetocontainapproximately75,000bacteria?Includeunits.

Part2:Now,HowAboutSomeLimits&Continuity:9)Findeachofthefollowinglimits,iftheyexist:

a)3 2

4

2 7 4lim4x

x x xx®

- --

b)9

3lim9x

xx®

--

c)2

1

2 5lim1x

x xx®

- -+

d) lim"→S

"#%A&$"%A"#

e)3

2

8lim2x

xx®-

++

f)lim"→T

"U$'&"#%)")"

g)limV→T

WX2 YU$V %WX2YU

V?

10)

11)If𝑓 𝑥 =𝑙𝑛𝑥𝑓𝑜𝑟0 < 𝑥 ≤ 2𝑥&𝑙𝑛2𝑓𝑜𝑟2 < 𝑥 ≤ 4,then lim"→& 𝑓(𝑥)is

(A)𝑙𝑛2(B)𝑙𝑛8(C)𝑙𝑛16(D)4(E)nonexistent12)13)

14)15)16)17)Findthevlaueof𝑘suchthatthefollowingfunctioniscontinuousforallrealnumbers

𝑓 𝑥 = 𝑘𝑥 − 1, 𝑥 < 2𝑘𝑥&, 𝑥 ≥ 2

(A)1(B)'

&(C)−'

((D)−'

&(E)noneofthese

18)

Part3:Ready,Set,Derive:19)Supposethatthefunctions𝑓𝑎𝑛𝑑𝑔andtheirderivativeswithrespectto𝑥havethefollowingvaluesatthegivenvaluesof𝑥.Findthederivativeof𝑓 𝑔 𝑥 at𝑥 = 4.

𝑥 𝑓 𝑥 𝑔 𝑥 𝑓a(𝑥) 𝑔a 𝑥

3 1 4 8 7

4 -3 3 5 -4

20)Findthederivativeforeachofthefollowing:a)𝑓 𝑥 = ln 𝑒&" b)𝑓 𝑥 = 4 + 𝑠𝑖𝑛10𝑥c)𝑓 𝑥 = 𝑥&𝑒"#

d)𝑓 𝑥 = 3𝑥−22𝑥+3e)𝑓 𝑥 = 3𝑠𝑖𝑛%'(5𝑥A)f)𝑓 𝑥 = "

b%!"

g)𝑓 𝑥 = 3𝑥&𝑠𝑒𝑐!𝑥h)𝑓 𝑥 = 3𝑥𝑐𝑠𝑐𝑥 + 𝑥𝑡𝑎𝑛𝑥

21)Findcd

c": 5𝑥! − 4𝑥𝑦 − 2𝑦& = 1

22)What’stheinstantaneousrateofchangeof𝑦 = ln(𝑐𝑜𝑠𝑥)atthepoint𝑥 = <

(

23)

24)Thedistanceofaparticlefromitsinitialpositionisgivenby ,wheresisfeetand

tisminutes.Findthevelocityatt=1minuteinappropriateunits.25)26)Aparticlemovesalongthe𝑥 − 𝑎𝑥𝑖𝑠sothatattime𝑡itspositionisgivenby𝑥 𝑡 = 𝑡! − 9𝑡& + 24𝑡for𝑡 ≥ 0.

a) Findthevelocityattime𝑡.b) Findtheaccelerationattime𝑡.c) Forwhatvaluesof𝑡doestheparticlechangedirection?d) Forwhatvaluesof𝑡isthespeeddecreasing?

)1(95)(+

+-=t

tts

27)Ahot-airballoonrisingstraightupfromaslevelfieldistrackedbyarangefinder500feetfromthelift-offpoint.Atthemomenttherangefinder’selevationangleis<

A,theangleisincreasingatthe

rateof0.14radiansperminute.Howfastistheballoonrisingatthatmoment?28)Thefigureaboveshowsthegraphof𝑓a,thederivativeofafunction𝑓.Thedomainofthefunctionisthesetofall𝑥suchthat−3 ≤ 𝑥 ≤ 3.

a)Forwhatvalue(s)of𝑥does𝑓havehorizontaltangents?

b)Onwhatintervalsisthegraphof𝑓increasing/decreasing?

c)Forwhatvaluesof𝑥,−3 ≤ 𝑥 ≤ 3,does𝑓havearelativemaximum?Arelativeminimum?Justifyyouranswer.

d)Forwhatvaluesof𝑥isthegraphof𝑓concaveup?Justifyyouranswer.

29)Findtheaveragerateofchangeofthefunctionoverthegiveninterval.

𝑓 𝑥 = 𝑥& + 2𝑥,[1,7]30)Attimet,thepositionofabodymovingalongthes-axisis𝑠 = 𝑡! − 27𝑡& + 240𝑡m.Findthebody’saccelerationeachtimethevelocityiszero.31)Findallcriticalvaluesforthefollowingfunction.Determineifthefunctionhasalocalmin,localmax,orneitheratitscriticalvalues.Explainyourreasoning.

𝑓 𝑥 =4 + 𝑥&

𝑥

Part4:Now,Let’sIntegrate!32)Findthegeneralsolutionforeachofthefollowing:a) 4𝑥5 − 3 𝑥 + 7

𝑥 𝑑𝑥b) (1 −'&𝑒A" + 𝑐𝑜𝑠3𝑥)𝑑𝑥c) 𝑥& 𝑥 − 6 𝑑𝑥

33)Ifcd

cO= j

O− '

OU+ 11and𝑦 = − '

&when𝑡 = 1,thenfindtheparticularsolutionforthe

differentialequation.

34)Whatareallthevaluesofkforwhich 02

5 =òk

dxx ?

35)Find c

c"𝑡 + 3k 𝑑𝑡&"

'

36)Evaluate: 2𝑐𝑜𝑠3𝑥𝑑𝑥YlT

37)Findtheaveragevalueof𝑓 𝑥 = 2𝑥 + 1overtheinterval 0,4 .38)Thefunctionfiscontinuousontheclosedinterval[1,9]andhasthevaluesgiveninthetable.

Usingthesubintervals[1,3],[3,6],and[6,9],whatisthevalueofthetrapezoidalapproximationof

ò9

1

)( dxxf ?

𝑥 1 3 6 9

𝑓(𝑥) 15 25 40 30

39)Thetablebelowprovidesdatapointsforthecontinuousfunction )(xhy = .

UsearightRiemannsumwith5subdivisionstoapproximatetheareaunderthecurveof)(xhy = ontheinterval[0,10].

𝑥 0 2 4 6 8 10

ℎ(𝑥) 9 25 30 16 25 32

40)Aparticlemovesalongthex-axissothat,atanytime 0³t ,itsaccelerationisgivenby 66)( += tta .Attime 0=t ,thevelocityoftheparticleis-9,anditspositionis-27.

(a) Find )(tv ,thevelocityoftheparticleatanytime 0³t .(b) Forwhatvaluesof 0³t istheparticlemovingtotheright?(c) Find )(tx ,thepositionoftheparticleatanytime 0³t .

41)Thegraphof𝑓shownconsistsofasemicircleandtwolinesegmentsontheinterval −2,6 .If𝑔 𝑥 = 𝑓 𝑡 𝑑𝑡"

& ,usethegraphof𝑓todothefollowing.

a) Find𝑔 −2 𝑎𝑛𝑑𝑔(4).

b) Findtheintervalswhere𝑔isdecreasing.Justifyyouranswer.

c) Findthepoint(s)ofinflectionof𝑔.Explainyourreasoning.

42)43)44)

45)46)47)48)

49)50)51)Findℎ′(2)ifℎ 𝑥 = 𝑔 𝑓 3𝑥 − 6𝑥,withthevaluesof𝑓 𝑥 𝑎𝑛𝑑𝑔(𝑥)areprovidedinthetablebelow.

52)

53)54)55)