Name: Period: AP CALCULUS AB -...
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Page1 Name: ____________________________________________________ Period: ____ AP CALCULUS AB: Summer Work 2020 For students to successfully complete the objectives of the AP Calculus curriculum, the student must demonstrate a high level of independence, capability, dedication, and effort. This summer packet is not intended to scare you, but is intended to help you maintain/improve your skills. This packet is a requirement for those entering the AP Calculus AB course and is due on the first day of class. At that time, it will be collected by your teacher and given a weight of a test grade. If it is not completed on the first day of class, you should consider yourself behind not only on your class grade (100 points), but also on the concepts necessary for success in Calculus. Complete as much of this packet on your own as you can, then get together with a friend or “google” the topic. There is a formula sheet at the end of this review. Feel free to use this sheet on this packet. Please know that you will not be supplied with a formula sheet for tests or quizzes during the Calculus course. Requirements: The following are guidelines for completing the summer work packet... • There are 80 questions (some with multiple parts) you must complete. You must show all of your work on the packet or, preferably, on a separate sheet of paper. • Be sure all problems are neatly organized and all writing is legible. • In the event that you are unsure how to perform functions on your calculator, you may need to read through your calculator manual to understand the necessary syntax or keystrokes. You must be familiar with certain built-in calculator functions such as finding values, intersection points, using tables, and finding zeros of a function. • I expect you to come in with certain understandings that are prerequisite to Calculus. A list of these topical understandings is below. Topical understandings within summer work... • Manipulate algebraic expressions involving exponents and radicals. • Manipulate algebraic fractions • Factor algebraic expressions • Solve equations through quadratics; complete the square in an algebraic expression • Solve simultaneous equations • Solve “word problems” (i.e. translate words into algebraic expressions) • Functions and graphs (rectangular coordinates) • Solve inequalities • Find roots of polynomials using synthetic division • Use binomial theorem • Manipulate complex numbers • Manipulate logarithmic expressions, graph logarithmic functions and solve logarithmic equations. • Solve exponential equations • Find equations of straight lines and conic sections • Determine inverse functions
Transcript of Name: Period: AP CALCULUS AB -...
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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)
