Serving the communities of Antioch, Lake Villa, Lindenhurst, and Old Mill Creek Community High School District 117, being a community of learners with a vision of excellence, is committed to providing an educational experience

that encourages all learners to develop to their fullest potential, to engage in lifelong learning, and to be responsible members of society.

Dear AP Calculus BC student, Hello and welcome to the wonderful world of AP Calculus! I am excited that you have elected to take an accelerated mathematics course such as AP Calculus BC and would like to welcome you to what will surely be a challenging yet rewarding year of mathematics. AP Calculus BC is intended for students who have a thorough knowledge of algebra, geometry, trigonometry and basic functions. The functions you must be familiar with include linear, polynomial, exponential, logarithmic, trigonometric, and piecewise-defined. In particular, before studying calculus, you must be familiar with the properties, algebra, and graphs of functions. Knowledge of the basic trigonometric identities is also essential, along with the values of the trigonometric functions of common angles. I would like to help you get a good start in AP Calculus BC. These problems are to be completed over the summer, and it is my expectation that all problems will be thoughtfully attempted, with all work and justification shown. Additionally, there is a significant amount of memorization expected for success on the AP exams. It is expected that students will return to school having mastered those procedural basics to permit more time to deeply understand the critically important conceptual aspects of the course. Also attached are 4 mostly new and skill-based calculus topics students are expected to master prior to the start of the school year. Please plan to devote approximately 2 hours of work to the review questions and at least 1 hour of work to each of the 4 new content topics. I look forward to supporting you in a successful and rewarding year in mathematics! Sincerely, Mrs. Cherestal

Anticipated workload: 6 hours Summer Packets are due Thursday, August 23, 2018

Summer Assignment Quiz (including a unit circle quiz) the same day

2

Algebra Key Concepts

1.Writetheequationofthelinethatpassesthroughthepoints(3,1)and(-5,-3)inpoint-slopeformthenconvertit

toslope-interceptform.2. Writeanequationofthelinethatistheperpendicularbisectorofthesegmentconnectinganx-interceptof2anda

y-interceptof6thenconvertittoslope-interceptform.3.Findthepointsofintersectionoftheliney=x+1andtheparabolay=x2–11usingalgebraicmethods.Showwork.

4.If𝑓 𝑥 = $%and𝑔 𝑥 = 𝑥' + 2𝑥find𝑓 𝑔 𝑥 and𝑔 𝑓 𝑥 .

Geometry Key Concepts

5.Iftheareaofarectangleis𝑥* − 3𝑥 − 4andtheheightoftherectangleis𝑥 + 1,findanexpressionforthelengthof

thebase.

3

6.Ifthevolumeofasphereis36π 𝑉 = 0'𝜋𝑟' ,findthesurfacearea 𝑆𝐴 = 4𝜋𝑟* .

7.Findthevolumeofacone(𝑉 = $'𝜋𝑟*ℎ)iftheratiooftheheighttotheradiusis3:1andtheconehasaheightof12

cm.8.Findtheareaofthelargertrapezoidif 9.Findtheshadedareaifthecirclesareconcentric,theshapesaresimilar. andthecircumferenceofthelargercircleis58π,and AB,whichistangenttothesmallercircle,is42units.

Advanced Algebra Key Concepts

Solveforallvaluesofx,wherexisarealnumber.Donotuseacalculatorunlessdirected.Leaveanswersexact.

10.1 + 2𝑥67 = 0 11.27*% = 9%;'

BA

C

65

74

4

12.log* 𝑥 − 5 = 3 13.𝑒% = cos 𝑥 *Useacalculator.Simplifywithoutacalculator:

14.ln 𝑒*7 15. 5𝑎E7 4𝑎

7E

16.logF(2𝑥) + 3 logF(𝑦) 17.sec arcsin N$O

Rewriteusingexponentproperties

18.(3𝑥*)0 19.%PEQP7

R

20. 3𝑥 21. 𝑏T F

22.'%%U 23. $*

%PV

5

Sketchaquickgraphofthefunctionlabelingkeyvalues(calculatorallowed)anddeterminewhetherthefunctionisevenorodd(explainwhatfeatureonthegraphindicatesevenv.odd):

24.𝑔 𝑥 = −3 cos %* 25.𝑓 𝑥 = 𝑥' − 𝑥

26.𝑓 𝑥 = 2 − 𝑥, 𝑥 ≤ 1𝑥*, 𝑥 > 1

a.Graphthefunction.Labelthecoordinatesofatleastthreepointsoneachgraph. b.Domain: c.Range: d.𝑓 2 =

e.𝑓 𝑓 −7 =Grapheachfunction.Labelanyverticalandhorizontalasymptotes,holes,xandy-intercepts:27.ℎ 𝑥 = *%[0

%E;%;F 28.𝑘 𝑥 = ln(𝑥 + 2)

Domain: Domain:Range: Range:

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10–11 x

1234567891011

–1–2–3–4–5–6–7–8–9–10–11

y

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10–11 x

1234567891011

–1–2–3–4–5–6–7–8–9–10–11

y

6

29. Fillouttheblankunitcircleentirely.Besuretoincludebothdegreesandradiansforeachangle.

30.cot O^F

31.csc N^'

32.sin '^0

33.sin;$(− '*)

34.csc;$(𝜋) 35.sec;$ − 2

7

Expand(noneedtoevaluate):

36.2! 37.3! 38.'!*! 39.7!

40.𝑛! 41. 𝑛 + 1 ! 42.(𝑛 + 2)! 43. a[$ !a!

44.Functions𝒇 𝒙 and𝒈(𝒙)aredescribednumericallybelow.Evaluatethefollowing:

A.)𝑓(2) B.)𝑔(−1) C.)𝑓 𝑔 4 D.)𝑓 𝑓 0

E.)𝑔 𝑓 4 F.)𝑓;$ 4 G.)𝑓 𝑔;$ 3 H.)𝑔;$ 𝑓;$ 7

45.Shownbelowarefunctions𝒇(𝒙)and𝒈(𝒙)describedgraphically.Evaluatethefollowing:

A.)𝑓(7) B.)𝑔(3) C.)𝑔 𝑓 3

D.)𝑓 𝑓 6 E.)𝑓;$(8) F.)𝑔;$ 4

G.)𝑔 𝑓;$ 3 H.)𝑓 𝑔;$ 1

𝑥 -1 0 2 4 7

𝑓(𝑥) -5 -1 4 7 8

𝑔(𝑥) 3 ½ 1/6 2 -6

8

THEFOLLOWINGPAGESCONTAIN

NEWCONTENTTOMASTERBEFORESCHOOLSTARTSJ

THETOPICSONTHEFOLLOWINGPAGESAREMOSTLYNEW.HONESTCOLLABORATIONWITHFRIENDSISENCOURAGED.

PLEASECONSULTTHEVIDEOLINKPRIORTOASKINGFORHELP.

WITHTHATSAID,PLEASEASKFORHELP.

CONTACTMRS.CHERESTALTHROUGHEMAIL:

Asra.cherestal@chsd117.org

DON’TPROCRASTINATE!

BECAUSETHECURRICULUMOFAPCOURSESISFAIRLYUNIVERSAL,EXCELLENTRESOURCESANDVIDEOSAREAVAILABLEONLINE.

I’DSUGGESTSTARTINGHERE:http://patrickjmt.com/#calculus

EVERYSINGLECALCULUSSTUDENTWILLNEEDHELPATSOMEPOINTTHISYEAR–LETSGETOFFONTHERIGHTFOOT

ANDASKFORHELPEARLYANDOFTEN!

9

Concept #1 – LimitsLearn– 1.Howtoevaluateone-sidedlimits 2.Howtoevaluatetwo-sidedlimits 3.Howtoevaluatelimitsatinfinity 4.Whenalimitequalsinfinityv.negativeinfinityv.doesnotexist(hint:notthesamething)Watch–https://www.youtube.com/watch?v=zpcBkRpqHqQ,http://patrickjmt.com/topic/limits/page/2/Try–Limithints–trythefollowingapproachesinorder:

1.“Plugitin,plugitin”–substitutethexvalueintothefunction.Ifthesolutionisarealnumber,congrats,you’redone.Ifitisg

g,∞∞, 0 ∗∞andotherconfusingformscalledindeterminate

(http://en.wikipedia.org/wiki/Indeterminate_form).ItdoesNOTequal0!Itmeansthevaluecannotbedeterminedwithoutfurtheralgebraicanalysis.2.Ifsubstitutingthex-valueyieldsanindeterminatesolution,tryoneofthealgebraicstrategiesbelowdependingonthefunctionform:

a.Rationalfunctions–factorthenumeratoranddenominator,simplify,andsubstituteagain.b.Compoundrationalexpressions–worktowardscommondenominatorsforfractionsbeingaddedorsubtractedthenreferto(a).c.Radicalexpressions–multiplybytheconjugateoftheradicalexpression(i.e. 𝑎 + 𝑏 ℎ𝑎𝑠𝑎𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒𝑜𝑓( 𝑎 − 𝑏)).

3.Analyzethelimitgraphically.0.Sketchagraphof𝑓 𝑥 = opq %

%.

a.Findlim

%→g𝑓(𝑥)fromthegraph. b.Find lim

%→t𝑓(𝑥)fromthegraph.

c.Memorizeeitherthetwolimitsaboveorthegraphof𝑓 𝑥 above.

d.lim%→g

opq *%0%

(Graphandmakelogicalconclusionaboutfutureopq u%v%

limits.Evaluateeachofthefollowinglimitswithoutagraphingcalculatorusingthehintsabove:

1. lim%→;*

𝑥' − 2𝑥* + 1 2. lim%→;*

%E[$'%E;*%[w

3.lim%→g

6Exy;

6E

% 4. lim

%→t*%E['w%E[O

5. lim%→t

%U[%E

$*%7[$*N 6. lim

%→;t'%E;*%;$%E['%;0

10

7. lim%→;t

%[opq %%[z{o %

8.lim%→$

'%E;*%;$%E['%;0

9.lim|→w

|[0;'|;w

10. lim|→t

'|[*w|E[$

+ 4{Useagraph}11.lim

%→g$% 12.lim

%→g− $%E

13.Giventhefunctionf x whosegraphisshownbelow(ticksmarksinpicturehavescale1): a) lim

%→'x𝑓 𝑥 = e)𝑓 3 =

b) lim

%→'P𝑓 𝑥 = f)𝑓 −1 =

d) lim

%→;$P𝑓 𝑥 = g) lim

%→;$𝑓 𝑥 =

e) lim

%→;$x𝑓 𝑥 = h)lim

%→'𝑓 𝑥 =

14.Sketchagraphofafunctionℎ(𝑥)thathasthefollowingproperties: a. lim

%→;tℎ 𝑥 = 0

b. lim

%→;*Pℎ 𝑥 = −∞

c. lim

%→;*xℎ 𝑥 = ∞

d.lim

%→gℎ 𝑥 = 0

e. lim

%→tℎ 𝑥 = 1

15.Sketchapossiblegraphforafunctionfthathasthestatedproperties.f(-2)exists,thelimitasxapproaches-2

exists,fisnotcontinuousatx=-2,andthelimitasxapproaches1doesnotexist.

11

Concept #2 – Definition of Derivative Learn– 1.Whatadefinitionofderivativerepresentswithrespecttorateofchangeandthefunction 2.Howtoapplythestandarddefinitionofderivative lim

�→g� %[� ;� %

%[� ;%orotherdefinitions

lim�→g

� %[� ;� %;�%[� ; %;�

, 𝑒𝑡𝑐.

3.HowaderivativecomparestoanaveragerateofchangeWatch–http://patrickjmt.com/understanding-the-definition-of-the-derivative/,http://patrickjmt.com/the-difference-quotient-example-2/Try–Usethedefinitionofderivative𝑓� 𝑥 = lim

�→g� %[� ;� %

%[� ;%todetermine𝑓′(𝑥)foreachofthefunctionsbelow.Use𝑓′(𝑥)

tofindtheslopeofthefunctionatthex-valuegiven,thenwriteatangentlineinpoint-slopeform.1.𝑓 𝑥 = 3𝑥 − 1(hint:ifyouunderstandwhataderivativeis,youdon’tevenneedtousethelimitforparta) a.Find𝑓′(𝑥). b.Find𝑓′(3). c.Findtheequationofatangentlineatx=3. d.Sketch𝑓 𝑥 andthetangentlineatx=3.2.𝑓 𝑥 = 𝑥* a.Find𝑓′(𝑥). b.Find𝑓′(−2). c.Findtheequationofatangentlineatx=-2. d.Sketch𝑓 𝑥 andthetangentlineatx=-2.

12

3.𝑓 𝑥 = 2𝑥* − 5𝑥 a.Find𝑓′(𝑥). b.Find𝑓′(1). c.Findtheequationofatangentlineatx=1. d.Sketch𝑓 𝑥 andthetangentlineatx=1.4.𝑓 𝑥 = 2𝑥'(hint: 𝑎 + 𝑏 ' = 𝑎' + 3𝑎*𝑏 + 3𝑎𝑏* + 𝑏')

a.Find𝑓′(𝑥). b.Find𝑓′(−1). c.Findtheequationofatangentlineatx=-1. d.Sketch𝑓 𝑥 andthetangentlineatx=-1.5.Carefullycomparetheoriginalfunctionsf(x)ineachoftheproblemsabovetothederivativesf’(x).Thereisafairly

simpleandincrediblypowerfulpatterncalledthePowerRule.Trytodescribeitin1-2sentencesbelow.

13

Concept #3 – Power Rule Learn– 1.Howtoquicklydeterminethederivativeofapolynomialfunction. 2.HowtoapplythePowerRuleforsimplerationalandradicalfunctions.Watch–https://www.youtube.com/watch?v=2CVCFkgsuNc,http://youtu.be/WrW5wVgZ3RsTry–1.Onceyou’veidentifiedthePowerRule,trytowriteitintheboxbelow,thefindf’(x)foreachfunctionbelowbyapplyingthePowerRule.

a.𝑓 𝑥 = 4𝑥 − 2 b.𝑓 𝑥 = 4𝑥* − 7𝑥 + 2

c.𝑓 𝑥 = 4𝑥* − 7𝑥 + 200000000 d.𝑓 𝑥 = 𝜋𝑥$g

e.𝑓 𝑥 = 𝑥w + 𝑥0 + 𝑥' + 𝑥* + 𝑥 + 1 f.𝑓 𝑥 = $%(hint:express$

%intheform𝑥ufirst)

g.𝑓 𝑥 = w%7 h.𝑓 𝑥 = 𝑥

i.𝑓 𝑥 = 𝑥*7 j.𝑓 𝑥 = $%7

2. 𝑓�� 𝑥 representsthe“secondderivative”ofthefunctionf(x)–thederivativeofthederivative.Similarly,𝑓′′′(𝑥)representsthethirdderivative,andforthefourthderivativeandhigher(sincetheapostropheswouldgetabitridiculous)derivativesareexpressedas𝑓 0 (𝑥),etc.

Find𝑓�� 𝑥 , 𝑓��� 𝑥 , 𝑓 0 𝑥 , 𝑓 w 𝑥 , 𝑎𝑛𝑑𝑓 F (𝑥)forthefunctionin(e)above.

PowerRule

If𝑓(𝑥) = 𝑥a,then𝑓�(𝑥) =

14

Forthethreefunctionsbelow,findtheaveragerateofchange(ARoC=� v ;� uv;u

)andinstantaneousrateofchange

(𝑓� 𝑥 usingthePowerRule),thenuse𝑓′(𝑥)towriteatangentlineandfindwheretheslopehasaparticularvalue.

3.𝑏 𝑥 = 5𝑥* − 2𝑥

AverageRoCover[0,2]? InstantaneousRoCexpression: X-valuewhereslopeis-12?

Tangentline@x=0?

4.𝑒 𝑥 = 𝑥' − 5𝑥

AverageRoCover[-3,3]? InstantaneousRoCexpression: X-valuewhereslopeis7?

Tangentline@x=1?

5.𝑓 𝑥 = 0%+ 3

AverageRoCover[1,2]? InstantaneousRoCexpression: X-valuewhereslopeis-1?

Tangentline@x=4?

6.Findthepoint(s)onthecurve𝑦 = 2𝑥' − 3𝑥* − 12𝑥wherethetangentlineishorizontal.

15

Concept #4 – Continuity and Differentiability Learn– 1.Howtodeterminewhereafunctioniscontinuousanddifferentiable

2.Howtosolveforamissingvalueinapiecewisefunctiongiventhatthefunctioniscontinuousanddifferentiable

3.WhattypesofdiscontinuitiesexistandwhatcausesfunctionstonotbedifferentiableWatch–http://youtu.be/YOuiXpLqDr0,http://youtu.be/VUEM6vWJvE4,http://youtu.be/Ei9p_h0mL_wTry–DefinitionofContinuity:Thefunctionfiscontinuousatsomepointcofitsdomainifthelimitoff(x)asxapproachescthroughthedomain

offexistsandisequaltof(c).Inmathematicalnotation,thisiswrittenas Indetailthismeansthreeconditions:first,fhastobedefinedatc.Second,thelimitofthatequationhastoexist.Third,thevalueofthislimitmustequalf(c).Thefunctionfissaidtobecontinuousifitiscontinuousateverypointofitsdomain.Primarydiscontinuitiesarepoint(hole),jump,infinite,andoscillatingdiscontinuities.

DefinitionofDifferentiability:Incalculus,adifferentiablefunctionofonerealvariableisafunctionwhosederivativeexistsateachpointinitsdomain.Asaresult,thegraphofadifferentiablefunctionmusthaveanon-verticaltangentlineateachpointinitsdomain,berelativelysmooth,andcannotcontainanydiscontinuities,corners,verticaltangents,orcusps.

1.FindthevalueofAsothatthefunctioniscontinuous.(Hint:Thetwopartsneedtohavethesamey-valueatthex-valuewheretheywould“meet”.)

𝑔 𝑥 = 5 − 𝐴𝑥*,𝑥 ≤ 14 + 3𝑥,𝑥 > 1

2.Determinethelocation(s)whereℎ(𝑥)isnotdifferentiableandwhatfeaturemakesitnotdifferentiable.

ℎ 𝑥 =1,𝑥 < −1𝑥* ,− 1 ≤ 𝑥 ≤ 0𝑥 + 1,𝑥 > 0

16

4. Drawasketchofagraphthatmeetsthefollowingrequirements.Thelimitdoesnotexistasxapproaches-2.Thefunctionalvalueatx=3isequaltothelimitoffasxapproaches3butthefunctionisnotdifferentiableatx=3.Thefunctionincreaseswithoutboundasxdecreaseswithoutbound.

5.DeterminethevaluesofAandBthatmakethefunctioncontinuousanddifferentiableatx=-2:

𝑔 𝑥 = −𝑥* + 8𝑥 − 7, 𝑥 ≤ −2𝐴𝑥 + 𝐵, 𝑥 > −2

6.Let𝑓 𝑥 = 𝑥 + 1 .Whichofthefollowingstatementsaretrueaboutf? I.fiscontinuousat𝑥 = −1 II.fisdifferentiableat𝑥 = −1 III.fhasacornerat𝑥 = −1 (A)Ionly (B)IIonly (C)IIIonly (D)IandIIIonly (E)IandIIonly7.Whichofthefollowingistrueaboutthegraphof𝑓 𝑥 = 𝑥*/wat𝑥 = 0? (A)Ithasacorner. (B)Ithasacusp. (C)Ithasaverticaltangent. (D)Ithasadiscontinuity. (E)𝑓 0 doesnotexist8.Sketchagraphofacontinuousfunction𝑏(𝑥)

if𝑏 3 = −1and:𝑏� 𝑥 =−1, 𝑥 < −21, −2 < 𝑥 < 1−2, 1 < 𝑥

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y