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⎛⎝⎜

⎞⎠⎟