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MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page1
Name:______________________________________________________________
CalculusSummerAssignmentDueDate:ThebeginningofclassonSeptember8,2017.
Thepurposeofthisassignmentistohaveyoupracticethemathematicalskillsnecessarytobesuccessful inCalculus. All of the skills covered in thispacket are fromPre-Calculus,Algebra2, andAlgebra1.Thematerialcoveredis fromourdistrictapproved,Pre-calculus,textbook.Ifyouneedto,youmayusereferencematerialstorefreshyourmemory(oldnotes,textbooks,onlineresources,etc.).While graphing calculators will be used during a few tests and quizzes, the majority of in classassessmentsarenon-calculator.Youareencouragedtolearnhowtobecalculator-independent.Attheendofthispage,therearelinkstosomesuggestedonlinecalculators.
CalculusisafastpacedcoursethatistaughtatthecollegeleveltoprepareyouforAPCalculus.Thereisalotofmaterialinthecurriculumthatmustbecoveredbeforetheendoftheyear.Therefore,wecannotspendalotofclasstimere-teachingprerequisiteskills. Thisiswhyyouhavethispacket.SpendsometimewithitandmakesureyouareclearoneverythingcoveredinthispacketsothatyouwillbesuccessfulinCalculus. Ofcourse,youarealwayswelcomedtoseekhelpfromyourteacherifnecessary.
This assignmentwill be collected and graded as your first test, the last class day of the firstweekofschool.Besuretoshowallappropriateworktosupportyouranswers.Inaddition,theremaybe a quiz on thismaterial during the first quarter.All questionsmust be completewith the correctwork.YoumustreturninSeptemberknowinghowtodoallthematerialinthispacket.
Forassistancewiththepacketyoumaycontactmeatgnaem@roselleschools.org.Emailsmaytakea fewdaysduring summer for a response. Pleasebe specific inyouremail forwhatyouneedassistancewith,includethesectionandthequestionnumberaswell.Foreachquestioninthepacket,there is an example in the separate tutorial packet. The tutorial packet is posted online on ourschool’swebsite.Referencetothecorrespondingsectionislistedundereachquestion. CalculatorsLinksOnline Calculator
https://www.desmos.com/calculator
https://mathway.com/graph
http://www.emathhelp.net/calculators/calculus-
1/online-graphing-calculator/
Emulator for Download
https://wabbit.codeplex.com/
http://lpg.ticalc.org/prj_tilem/download.html
GoodLuck!
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page2
1) Writeanequationofalinethroughthepoint(a)paralleltothegivenlineand(b)perpendicularto
thegivenline:(TutorialSectionI)
a)Point:(1,5)Line:6x–2y=8
b)Point:(–2,2)Line:3x+5y=8
2) Findtheslopeandy-interceptoftheline:
(TutorialSectionI)a)2x–3y=12
b)4x+y=1
c)y=2
d)x=–51
3) Findtheequationofalineinslopeinterceptform:
(TutorialSectionI)a)Contains(1,2)and(–2,4)
b)Contains(2,1)andm=4
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page3
c)Contains(1,7)andm=0
d)Contains(6,5)andmisundefined
4) Writetheequationofthefollowinglinesinpointslopeform:
(TutorialSectionI)a)Contains(–1,4)and(3,8)
b)Passesthrough(6,2)andhaday–interceptof5
5) Sketchagraphoftheequation:
(TutorialSectionI)a)y=–3x+2 b)x=4–y
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page4
c)y–1=3x+12
d)x=7
6) Findtheinterceptsforeachofthefollowing.
(TutorialSectionII)a) 3 2y x= − + b) 3 2y x= +
c)2
2
3(3 1)x xyx+=+
d)
2 3 4y x x= −
7) Findthepoint(s)ofintersectionofthegraphsforthegivenequations.
(TutorialSectionIII)
a)x + y = 84x − y = 7
⎧⎨⎩
b)x2 + y = 6x + y = 4
⎧⎨⎩
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page5
c)
x2 − 4y2 − 20x − 64y −172 = 016x2 + 4y2 − 320x + 64y +1600 = 0
⎧⎨⎪
⎩⎪
8) Letf(x)=3x+2andg(x)=1+x2findeachofthefollowing:
(TutorialSectionIV)a)f(g(0))=
b)g(g(2))=
c)g(f(x))= d)f(g(x))=
9) If𝒇 𝒙 = 𝒙𝟐,𝒈 𝒙 = 𝟐𝒙 − 𝟏.𝒂𝒏𝒅 𝒉 𝒙 = 𝟐𝒙, 𝒇𝒊𝒏𝒅 𝒕𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈:
(TutorialSectionIV)a)f(h(-1))=
10) Forf(x)=8x–3,find
(TutorialSectionV)a)! !!! !!(!)
!=
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page6
11) Forf(x)=x2,find.(TutorialSectionIV)
12) Findthedomainandrangeforeachfunctiongiveyouranswerusingintervalnotation:
(TutorialSectionIV)a) 2( ) 9h x x= − b)h(x)=sinx
c) 2( )2 3
f xx
=+ d)
2 1, 0( )
2 2, 0x x
f xx x+ <⎧
= ⎨ + ≥⎩
e) 𝑦 = !!!!!!!!
f) 𝑦 = !!!!!!!!
g)𝑦 = !!!!!!!!!!!!!!"
a)! !!! !!(!)!
=
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page7
13) Testforsymmetrywithrespecttoeachaxisandtheorigin.(TutorialSectionV)
a)𝑦 = 𝑥 𝑥 + 2
b)𝑦 = 6− 𝑥
c) 𝑦 = !!!!!
d)𝑦 = 𝑥! − 𝑥
14) Show work to determine if the relation is even, odd, or neither:
(TutorialSectionV) a)𝑓 𝑥 = 2𝑥! − 7 b)𝑓 𝑥 = −4𝑥! − 2𝑥
c) 𝑓 𝑥 = 4𝑥! − 4𝑥 + 4
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page8
15) For each function below list all holes, vertical asymptotes and x-intercepts:(TutorialSectionVI)
a) f (x) = (x − 3)(x + 2)
(x − 3)(2x +1) b)
y = x2 −1
2x2 + x −1
c) 82
3212)( 2
23
−−+−=xx
xxxxf d) 23
149)( 2
2
+++−=xxxxxg
16) Find the inverse for each function.
(TutorialSectionVII) a) f (x) = 2x +1 b) f (x) = x2
3
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page9
17) Provefandgareinversesofeachother.(TutorialSectionVII)
a)f (x) = x3
2g(x) = 2x3
b) f (x) = 9 − x2, x ≥ 0 g(x) = 9 − x
18) Solve each equation:
(TutorialSectionVIII) a)7𝑥! − 3𝑥 = 0
b)4𝑥 𝑥 − 2 − 5𝑥 𝑥 − 1 = 2
c) 𝑥! + 6𝑥 + 4 = 0
d) 2𝑥! − 3𝑥 + 3 = 0
e) 2𝑥! − 𝑥 + 2 𝑥 − 3 = 12 f) 𝑥 + !!= !"
!
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page10
19) Solve each inequality:
(TutorialSectionVIII) a)𝑥! − 16 > 0 b)𝑥! + 6𝑥 − 16 > 0
c) 𝑥! − 3𝑥 ≥ 10
d) 2𝑥! + 4𝑥 ≤ 3
e) 𝑥! + 4𝑥! − 𝑥 ≥ 4 f) 2 sin! 𝑥 ≥ sin 𝑥, 0 ≤ 𝑥 < 2𝜋
20) Solveforx:
(TutorialSectionVIII)a)!
!− !
!= !
!
b)𝑥 + !!= 5
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page11
c) !!!!− !!!
!= 1
d)
52
31
15=+
xx
21) Write the following absolute value expressions as piecewise expressions (by remove the absolute value):
(TutorialSectionIX) a)𝑦 = 2𝑥 − 4
b) 𝑦 = 6 + 2𝑥 + 1
c) 𝑦 = 4𝑥 + 1 + 2𝑥 − 3
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page12
22) Solvethefollowingabsolutevalueinequalities:(TutorialSectionIX)
a) 𝑥 − 3 > 12 b) 𝑥 − 3 ≤ 4
c) 10𝑥 + 8 > 2 d) 3𝑥 − 4 > −2
e) 𝑥 − 6 > −8
23) Writewithoutfractionalexponents:
(TutorialSectionX)a) 3/12xy = b) 4/12 )16()( xxf =
c)4/33/127 xy = d) 9
!! =
e) 64!! =
f) 8
!! =
g) 27!! =
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page13
24) Writewithpositiveexponents:(TutorialSectionX)
a) 32)( −= xxf b) 22 )4( −= xy
c) 𝑦 = !!!!!
!! d) 𝑓 𝑥 = !!! !!
!!!! !!
25) Factorthensimplify:
(TutorialSectionX)a) f (x) = 4x−3 + 2x −18x−2 b) f (x) = 5x2 (x − 2)−1/2 + (x − 2)1/23x
c) f (x) = 6x(2x −1)−1 − 4(2x −1)
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page14
26) Simplifyrationalexpressions:(TutorialSectionX)
a) f (x) = (4x2 )3
2x
b) y = (2x +1)(x −3)2
(x −3)4(2x −1)
c) f (x) = 16x2 −8x +1
4x2 +3x −1
d) y = x2 − 25x2 −10x + 25
27) Simplify:
(TutorialSectionXI)a)!
!+ !
!= b)!
!− !
!=
c)!!∙ !!= d)
!!!!=
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page15
e) !!!!!!!
= f) !!!!"!!!"!!!!!!!"
=
g) !!!
!!
!!!= h) !
!!!− !
!=
i) !!!!!!!!
= j) !!!!!!!!
=
k) !!!!!
=l)
9 3xx+ − =
m)x h xh
+ − =
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page16
28) Expressasasinglelogarithm:
(TutorialSectionXII)a)3 ln 𝑥 + 2 ln 𝑦 − 4 ln 𝑧 = b)3 ln 𝑥 = 1
c)Solveforx: 𝑒!!! = 7
29) Graphtwocompleteperiodsofthefunction.
(TutorialSectionXIV)a)
c)
c)
d)f (x) = − cos x −π4
#$%
&'(
f (x) = 5sin x f (x) = sin2x
f (x) = cos x − 3
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page17
e) f (x) = cos(2x +π )− 2
f)f x( ) = 2sin x +π( )−1
g) f x( ) = − tan x2+π2
"
#$
%
&'
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page18
30) Performthefollowingtransformationsbasedonthegivendiagram.
(TutorialSectionXV)
a) ( ) ( ) 12 −−= xfxh
b) ( ) ( ) 21 ++−= xfxk
c) ( ) ( ) 2+−= xfxj
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page19
31) Identifytheparentfunctionsoff,describethesequenceoftransformationsfromftoh.(TutorialSectionXVI)
a) ( ) ( ) 13 2 ++−= xxh b) ( ) 7−= xxh
32) Graph the following circle:
(TutorialSectionXVII) a)(x-3)2+(y+1)2=4
33) Answer the following questions:
(TutorialSectionXVII) a)Findthestandardform,center,andradiusofthefollowingcircles:x2+y2–4x+8y–5=0
b)Givetheequationofthecirclewhosecenteris(5,-3)andgoesthrough(2,5)
34) Factorthefollowing:
(TutorialSectionXVIII)a) 27x3+125 b) 5129 −x
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page20
35) Complete the following table: (TutorialSectionXX)
a)
θ in Degrees (Radians) Cosθ Sinθ Quadrant
0°
30°
45°
60°
90°
120°
135°
150°
180°
210°
225°
240°
270°
300°
315°
330°
360°
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page21
36) Findallsolutionstotheequations.Youshouldnotneedacalculator.(Hint:oneofthesehasNOsolution.)(TutorialSectionXX)
a) 4cos2 x − 4cos x = −1 b) 2sin2 x +3sin x +1= 0
c) sin2 x − 2sin x = 0 d)3sin x = 2cos
2 x
e) 2sin2 x +3sin x = 2 f) cos2x + 5cos x = 2 (hint:usedoubleangleidentity)
g) sin(cos x) =1 h) sin2 x − 2sin x −3= 0
MATHACHSCALCULUSSUMMERASSIGNMENTJUNE,2017Page22
37) Foreachofthefollowing,expressthevaluefor“y”inradians.(TutorialSectionXXI)
a) y = arcsin − 3
2
b) y = arccos −1( )
c) y = arctan(−1)
38) Foreachofthefollowinggivethevaluewithoutacalculator.
(TutorialSectionXXI)
a) tan arccos
23
⎛⎝⎜
⎞⎠⎟ c)
sec sin−1 12
13⎛⎝⎜
⎞⎠⎟
d) sin arctan12
5⎛⎝⎜
⎞⎠⎟ e)
sin sin−1 7
8⎛⎝⎜
⎞⎠⎟