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Name:____________________________________________________Period:____
APCALCULUSAB:SummerWork2020
ForstudentstosuccessfullycompletetheobjectivesoftheAPCalculuscurriculum,thestudentmustdemonstrateahighlevelofindependence,capability,dedication,andeffort.Thissummerpacketisnotintendedtoscareyou,butisintendedtohelpyoumaintain/improveyourskills.ThispacketisarequirementforthoseenteringtheAPCalculusABcourseandisdueonthefirstdayofclass.Atthattime,itwillbecollectedbyyourteacherandgivenaweightofatestgrade.Ifitisnotcompletedonthefirstdayofclass,youshouldconsideryourselfbehindnotonlyonyourclassgrade(100points),butalsoontheconceptsnecessaryforsuccessinCalculus.Completeasmuchofthispacketonyourownasyoucan,thengettogetherwithafriendor“google”thetopic.
Thereisaformulasheetattheendofthisreview.Feelfreetousethissheetonthispacket.PleaseknowthatyouwillnotbesuppliedwithaformulasheetfortestsorquizzesduringtheCalculuscourse.
Requirements:Thefollowingareguidelinesforcompletingthesummerworkpacket...
• Thereare80questions(somewithmultipleparts)youmustcomplete.Youmustshowallofyourworkonthepacketor,preferably,onaseparatesheetofpaper.
• Besureallproblemsareneatlyorganizedandallwritingislegible.• Intheeventthatyouareunsurehowtoperformfunctionsonyourcalculator,youmayneedto
readthroughyourcalculatormanualtounderstandthenecessarysyntaxorkeystrokes.Youmustbefamiliarwithcertainbuilt-incalculatorfunctionssuchasfindingvalues,intersectionpoints,usingtables,andfindingzerosofafunction.
• IexpectyoutocomeinwithcertainunderstandingsthatareprerequisitetoCalculus.Alistofthesetopicalunderstandingsisbelow.
Topicalunderstandingswithinsummerwork...
• Manipulatealgebraicexpressionsinvolvingexponentsandradicals.• Manipulatealgebraicfractions• Factoralgebraicexpressions• Solveequationsthroughquadratics;completethesquareinanalgebraicexpression• Solvesimultaneousequations• Solve“wordproblems”(i.e.translatewordsintoalgebraicexpressions)• Functionsandgraphs(rectangularcoordinates)• Solveinequalities• Findrootsofpolynomialsusingsyntheticdivision• Usebinomialtheorem• Manipulatecomplexnumbers• Manipulatelogarithmicexpressions,graphlogarithmicfunctionsandsolvelogarithmicequations.• Solveexponentialequations• Findequationsofstraightlinesandconicsections• Determineinversefunctions
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Trigonometryalsoplaysanimportantroleincalculusandisusedthroughoutthecourse.Thestudentmustknowtrigonometryinordertobesuccessfulinthecourse.Inparticular,thestudentshouldbefamiliarwiththefollowing.
• Fundamentaldefinitions• Basicidentities• Applicationofbasicidentitiestothesolutionsoftrigonometricequationsandprovingidentities• Graphingtrigonometricfunctions• Radianmeasure• Theinversetrigonometricfunctions• Domainandrangeoftrigonometricandinversetrigonometricfunctions
Finally,Isuggestnotwaitinguntilthelasttwoweeksofsummertobeginonthispacket.Ifyouspreaditout,youwillmostlikelyretaintheinformationmuchbetter.Onceagainthisisdueonthefirstdayofclass.Bestofluckandifyouhaveanyquestions,[email protected].
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SummerReviewPacketforStudentsEnteringCalculus(ABlevel)
ComplexFractionsWhensimplifyingcomplexfractions,multiplybyafractionequalto1whichhasanumeratoranddenominatorcomposedofthecommondenominatorofallthedenominatorsinthecomplexfraction.Example:
−𝟕 − 𝟔𝒙 + 𝟏𝟓
𝒙 + 𝟏=−𝟕 − 𝟔
𝒙 + 𝟏𝟓
𝒙 + 𝟏∙𝒙 + 𝟏𝒙 + 𝟏 =
−𝟕𝒙 − 𝟕 − 𝟔𝟓 =
−𝟕𝒙 − 𝟏𝟑𝟓
− 𝟐𝒙 +
𝟑𝒙𝒙 − 𝟒
𝟓 − 𝟏𝒙 − 𝟒
=−𝟐𝒙 +
𝟑𝒙𝒙 − 𝟒
𝟓 − 𝟏𝒙 − 𝟒
∙𝒙(𝒙 − 𝟒)𝒙(𝒙 − 𝟒) =
−𝟐(𝒙 − 𝟒) + 𝟑𝒙(𝒙)𝟓(𝒙)(𝒙 − 𝟒) − 𝟏(𝒙) =
−𝟐𝒙 + 𝟖 + 𝟑𝒙𝟐
𝟓𝒙𝟐 − 𝟐𝟎𝒙 − 𝒙
Simplifyeachofthefollowing.
1. mnopq rstr
2.uqmtv
wnmtr 3.
4.If𝑓(𝑥) = 3𝑥p + 5𝑥 + 1,whatis}(mn~)q}(m)
~?
2xx2 − 6x + 9
−1x +1
−8
x2 − 2x −3
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Exponential,LogarithmicandRationalExpressionsSimplifyeachofthefollowing,withoutacalculator.
5. 6. (Rationalizethedenominator)
7. 8.
9.ln1 10.
11. 12.
13.Rationalizethedenominator √�p�
�um�� 14.
15. 16.
17. 18.logo�(log10,000)w
5a23( ) 4a32( ) 4
3 + 5
2 log x −3( )+ log x + 2( )− 6 log x x − 4x2 −3x − 4
5− xx2 − 25
e 1+ln x( )
2x210x5
1x−15
1x2−125
log128 x
32 x + x
52 − x2( )
e3ln x
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FunctionsToevaluateafunctionforagivenvalue,simplyplugthevalueintothefunctionforx.Recall: read“fofgofx”meanstoplugtheinsidefunction(inthiscaseg(x))inforxintheoutsidefunction(inthiscase,f(x)).Example:Given findf(g(x)).
Let𝒇(𝒙) = −𝟓𝒙 + 𝟕and𝒈(𝒙) = 𝟑𝒙𝟐 + 𝟗.Findeach.19.𝑓(−12) =____________ 20. _____________ 21.𝑔(𝑡 + 2) =__________
22. __________ 23. ___________ 24. ______
FactorCompletely:25.16𝑥p − 8𝑥 + 1 26.2𝑥w − 5𝑥p − 18𝑥 + 4527.36𝑥p − 100𝑦p 28.10𝑥p + 13𝑥 − 3
f g( )(x) = f (g(x)) OR f [g(x)]
f (x) =2x2 +1 and g(x) = x − 4
f (g(x)) = f (x − 4)= 2(x − 4)2 +1= 2(x2 − 8x +16) +1= 2x2 −16x + 32 +1
f (g(x)) = 2x2 −16x + 33
g(−3) =
f g(−2)"# $% =
g f (m + 2)!" #$ = f (x + h) − f (x)
h=
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EquationofalineSlopeinterceptform: Verticalline:x=c(slopeisundefined)Point-slopeform: Horizontalline:y=c(slopeis0)29.Findtheequationofalineperpendicularto passingthroughthepoint(-16,71).30.Findtheequationofalinepassingthroughthepoints(-12,4)and(6,5).VerticalAsymptotesDeterminetheverticalasymptotesforthefunction.Setthedenominatorequaltozerotofindthex-valueforwhichthefunctionisundefined.Thatwillbetheverticalasymptote.
31. 32. 33.
HorizontalAsymptotesDeterminethehorizontalasymptotesusingthethreecasesbelow.CaseI.Degreeofthenumeratorislessthanthedegreeofthedenominator.Theasymptoteisy=0.CaseII.Degreeofthenumeratoristhesameasthedegreeofthedenominator.Theasymptoteistheratiooftheleadcoefficients.CaseIII.Degreeofthenumeratorisgreaterthanthedegreeofthedenominator.Thereisnohorizontalasymptote.Thefunctionincreaseswithoutbound.(Ifthedegreeofthenumeratorisexactly1morethanthedegreeofthedenominator,thenthereexistsaslantasymptote,whichisdeterminedbylongdivision.)DetermineallHorizontalAsymptotes.
34. 35. 36.
y = mx + b
y − y1 = m(x − x1)
2x −3y = 5
f (x) = 1x2
f (x) = x2
x2 − 4f (x) = 2 + x
x2 (1− x)
f (x) = x2 − 2x +1x3 + x − 7 f (x) = 5x
3 − 2x2 + 84x − 3x3 + 5
f (x) = 4x5
x2 − 7
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Solvethefollowingequationsorinequalities:Ifusingadecimalsolution,statethesolutiontothree(3)decimalplaces.37. 38.
39. 40.
41. 42.w
�− o
�m= o
m
43. 44. 45. 46.
4t3 −12t2 +8t = 0 3 x − 2 −8 = 8
log x + log x −3( ) =1 x − 53− x
> 0
4e2 x = 5
272 x = 9x−3 x +1( )2 x − 2( )+ x +1( ) x − 2( )2 = 0
ln3x + ln3= 3 log5 x +3( )− log5 x = 2
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TrigonometricEquations:Isolatethevariable,sketchareferencetriangle,findallthesolutionswithinthedomain .Usetrigidentities,ifneeded,torewritethetrigfunctions.(Seeformulasheetattheendofthepacket.)
47. 48.cos 𝑥 = sin 𝑥
49.cosw 𝑥 = cos 𝑥 50.sin 𝑥 − 2 sin 𝑥 cos 𝑥 = 051.3 cotp 𝑥 − 1 = 0 52.
0 ≤ x < 2π
sin x = − 12
4 cos2 x − 3 = 0
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Foreachofthefollowing,findthevaluewithoutacalculator.
53.arcsin �− √wp� 54. 55.
56.tan �cosqo �− p
w�� 57.csc �cosqo op
ow�
58. 59.
y = arccos −1( ) cos sin−1 1
2"
#$
%
&'
sin arctan12
5!
"#$
%& sin sin−1 7
8"
#$%
&'
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Withoutacalculator,determinetheexactvalueofeachexpression,ifitexists.
60. 61. 62.cot �w
63. 64.sec ��w 65.
66. 67. 68.
DomainandRange:Determinethedomainofeachfunction.Writeyouranswerinintervalnotation.69.𝑦 = √𝑥p + 4 Domain_______________________________
70. Domain_______________________________
71.𝑦 = √16 − 𝑥p Domain_______________________________ 72.𝑦 = ln(2 − 𝑥) Domain_______________________________ 73. Domain_______________________________ 74.𝑓(𝑥) = √wqm
mno Domain_______________________________
Sketchthegraphwithoutacalculator.
75.𝑓(𝑥) = �−𝑥p − 2 ≤ 𝑥 < 1−4𝑥 = 12𝑥 − 11 < 𝑥 ≤ 5
sin0 cos 7π6
sin π2
tan π2
sin 3π4
tan 7π4
sin−1 sin 7π6
"
#$
%
&'
g x( ) = x − 2x2 − 4
h x( ) = 2x +1
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MiscellaneousTopicsFromPastCourses
76.Thenumberofelkaftertyearsinastateparkismodeledbythefunction .
a.Whatwastheinitialpopulationofelk?
b.Whenwillthenumberofelkbe750?Roundyouranswertothenearestwholenumberandstatetheunits.
77.Usepolynomialdivisiontostatethequotientandremainderofpm
�q�mvn�mqumqp
.78.Useagraphingcalculatortosolvetheequationforx.Roundyouranswerto3decimalplaces. .
P t( ) = 12161+ 75e−0.03t
e2 x = 3x2
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FreeResponseQuestion:Answerthefollowingquestionscompletely.79.Thetemperatureoutsideahouseduringa24-hourperiodisgivenby
𝐹(𝑡) = 80 − 10 cos �𝜋𝑡12� ,for0 ≤ 𝑡 ≤ 24
whereF(t)ismeasuredindegreesFahrenheitandtismeasuredinhours.
a. SketchthegraphofFonthegridbelow.
b. Definetheamplitude,period,phaseshift,andverticalshiftofthefunctionF.
c. Withinthecontextoftheproblem,explainthemeaningofthepoint𝐹(6) = 80.
d. Findtheaveragetemperaturebetweent=6andt=14.Keeptheanswerinsimplifiedradicalform.
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80.Ericrideshisbiketoschoolalongastraightroadstartingfromhomeatt=0minutes.Forthefirst8minutes,Eric’svelocity,inmilesperminute,ismodeledbythepiecewiselinearfunctionwhosegraphisshownabove.
a. HowfastisEricridingatt=2minutes?Includeunitsinyouranswer.
b. Writeanequationfortheportionofthevelocityfunctionbetweent=2andt=4minutes.UseyourequationtofindEric’svelocityatt=3.8minutes.
c. Ericdecreaseshisvelocityfromt=8tot=12ataconstantratesothathisvelocityatt=12is0miles/minute.Onthegraph,extendthevelocityfunctiontoshowthisnewinformation.
d. Theareabetweenthevelocitygraphandthex-axisrepresentsthedistanceErictravelled.UseyourunderstandingofareaofpolygonstodeterminethetotaldistanceErictravelledin12minutes.
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cos2x = cos2 x − sin2 x
= 1− 2sin2 x
= 2cos2 x −1
FormulaSheet
ReciprocalIdentities:
QuotientIdentities:
PythagoreanIdentities: DoubleAngleIdentities:
Logarithms: isequivalentto Productproperty:
Quotientproperty:
Powerproperty: Propertyofequality: If ,thenm=n
Changeofbaseformula:
DerivativeofaFunction: Slopeofatangentlinetoacurveorthederivative:
Slope-interceptform: Point-slopeform: Standardform: Ax+By+C=0
csc x = 1sin x
sec x = 1cos x
cot x = 1tan x
tan x = sin xcos x
cot x = cos xsin x
sin2 x + cos2 x = 1 tan2 x +1 = sec2 x 1+ cot2 x = csc2 x
sin2x = 2sin x cos x
tan2x = 2 tan x1− tan2 x
y = loga x x = ay
logb mn = logb m + logb n
logbmn= logb m − logb n
logb mp = p logb m
logb m = logb n
loga n =logb nlogb a
limh→0
f (x + h) − f (x)h
y = mx + b
y − y1 = m(x − x1)
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