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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)