paul a foerster precalculus with trigonometry concepts and applications

Third Edition

P A U L A . F O E R S T E R

SOLUTIONS MANUAL

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Editors: Jocelyn Van Vliet, Elizabeth DeCarliMath Checker: Cavan FangProject Administrator: Tamar ChestnutProduction Editor: Angela ChenEditorial Production Supervisor: Kristin FerraioliCopyeditor: Margaret MooreProduction Director: Christine OsborneText Designers: Adriane Bosworth, Marilyn PerryArt Editor: Maya MelenchukTechnical Art: Lineworks, Inc., Interactive Composition Corporation, Saferock USA LLCCover Designer: Diana GhermannCover Photo Credit: Getty Images/moodboardComposition and Prepress: Saferock USA LLCPrinter: RR DonnelleyTextbook Product Manager: Elizabeth DeCarliExecutive Editor: Josephine NoahPublisher: Steven Rasmussen

2012 by Key Curriculum Press. All rights reserved.

Limited Reproduction PermissionThe publisher grants the teacher who purchases Precalculus with Trigonometry: Concepts and Applications Solutions Manual the right to reproduce material for use in his or her own classroom. Unauthorized copying of Precalculus with Trigonometry: Concepts and Applications Solutions Manual constitutes copyright infringement and is a violation of federal law.

Key Curriculum Press is a registered trademark of Key Curriculum Press. The Geometers Sketchpad, Sketchpad, Fathom, Fathom Dynamic Data, and the Fathom logo are registered trademarks of KCP Technologies. All other trademarks are held by their respective owners.

Key Curriculum Press1150 65th StreetEmeryville, CA [email protected]

Printed in the United States of America10 9 8 7 6 5 4 3 2 1 15 14 13 12 11ISBN 978-1-60440-058-8

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iii

Contents

Overview of Solutions Manual v

Chapter 1 Functions and Mathematical Models 1

Chapter 2 Properties of Elementary Functions 21

Chapter 3 Fitting Functions to Data 35

Chapter 4 Polynomial and Rational Functions 51

Chapter 5 Periodic Functions and Right Triangle Problems 79

Chapter 6 Applications of Trigonometric and Circular Functions 87

Chapter 7 Trigonometric Function Properties and Identities, and Parametric Functions 103

Chapter 8 Properties of Combined Sinusoids 119

Chapter 9 Triangle Trigonometry 131

Chapter 10 Conic Sections and Quadric Surfaces 145

Chapter 11 Polar Coordinates, Complex Numbers, and Moving Objects 169

Chapter 12 Three-Dimensional Vectors 183

Chapter 13 Matrix Transformations and Fractal Figures 199

Chapter 14 Probability, and Functions of a Random Variable 219

Chapter 15 Sequences and Series 231

Chapter 16 Introduction to Limits, Derivatives, and Integrals 241

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Overview of Solutions Manual

The Precalculus with Trigonometry: Concepts and Applications Solutions Manual contains one possible complete solution, including key steps and commentary where necessary, to each of the problems at the end of each section in the student text.

Solutions are presented in the form your students would be expected to use. Bear in mind, though, that there may be more than one way to solve any given problem using a correct method.

As in the student text, exact answers are displayed using the ellipsis format. When real-world approximations are required in the answer, exact calculations are used until the final answer is found, and then the appropriate rounding is indicated.

Where calculator programs are called for, sample programs and commentary are provided at www.keymath.com/precalc. The programs can be downloaded to TI-83, TI-84, and TI-Nspire calculators.

Solutions are not provided for journal entries. Student responses are highly individual and will vary from student to student.

v

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Problem Set 1-1 1. a. 20m;217.5m;itisbelowthetopofthecliff.

b.0.3s;03.8s;5.3s c.5m

d.Thereisonlyonealtitudeforanygiventime;somealtitudescorrespondtomorethanonetime.

e.Domain: 0 x 5.3;range:230y 25.

2. a.

300 450

10

5

20

35V (liters)

T (C)

Thisgraphalsoshowstheanswerforpartbbelow.

b.Answerswillvary.V (400)23,V (30)11,andV (T )50whenT2273.Absolutezeroisabout2273C.

c.Extrapolation:V (400)andTsuchthatV (T )50;interpolation:V (30).

d.Thereisonlyonevolumeforagiventemperature;yes,becausethereisonlyonetemperatureforagivenvolume.

e.Domain:x 2273,orwhatevernumberisfoundinpartb;range:y 0.(Strictinequalitiesareusedbecauseabsolutezeroisatheoreticallimitwhichcantbeattained,andthevolumecanneverequalzero.)

3. a. n B

0 150,000

12 145,995

24 141,744

36 137,230

48 132,438

60 127,350

72 121,948

84 116,213

96 110,125

108 103,661

120 96,798

b.ChangingTblto1showsthatthebalancebecomesnegativeattheendofmonth241,sothebalancewillbecome0duringmonth241.Inreality,thebalancewouldbepaidoffattheendofmonth241,butwithasmallerpayment,$3.04ratherthan$1074.64.(AfterstudyinglogarithmsinChapter2,studentswillalsobeabletosolvethisequationalgebraically.)

n B

235 5296.5

236 4248.3

237 3194.9

238 2136.3

239 1072.3

240 3.0438

241 21072

c.

100 200

100,000

n

B

d.False

100 200

100,000

x

y

e.Domain:0x 241,x is aninteger;range:0y 150,000.Thevaluesarecalculatedonlyatwhole-monthintervals.(Therangevaluesalsojumpfromonetothenext,butingeneraltheyarenotintegers.)

4. a.

Speed

Distance

b.0x 65ifyoustaywithinthespeedlimit.

c.AccordingtotheTexas Drivers Handbook, thedistancewouldbeabout240ft.

d.Policeconsiderthelengthoftheskidmarkstheindependentvariable.

Speed

Distance

Precalculus with Trigonometry: Solutions Manual Problem Set 1-1 1 2012 Key Curriculum Press

Chapter 1 Functions and Mathematical Models

PC3_SM_Ch01.indd 1 6/23/11 1:40:17 PM

5. Thisgraphassumesthattheelementheatsfromaroomtemperatureof72Ftonearlyamaximumtemperatureof350Finoneminute.

30 60

200

y

x

Domain: x 0 s;range: 72Fy 350F.

6. a. 1:graphically(andverbally); 2:numerically;3:algebraically; 4:verbally;5:verbally

b.1:graphicaltonumerical;2:numericaltographical,thengraphicaltonumericalfortheextrapolationandinterpolation;3:algebraictonumericalandalgebraictographical;4:verbaltographical;5:verbaltographical

Problem Set 1-2 1. a.

4 8

10

20

x

y

b.3f (x )23

c.Linear

d.Answerswillvary;e.g.,thecost(inthousandsofdollars)ofmanufacturingxitemsifeachitemcosts$2000tomanufactureandthereisa$3000start-up.

2. a.

2 4

10

y

x

b.0f (x )12.8

c.Power

d.Answerswillvary;e.g.,theweightofananimalbasedononeofitslineardimensions.

3. a.

4 8

20

40

y

x

b.g (x )1.2

c. Inversevariation

d.Answerswillvary;e.g.,thetimeittakestogo12miatxmi/h.

4. a.

20

40

y

x4 4

b.0.3888h (x )64.3 c.Exponential

d.Answerswillvary;e.g.,thenumberofbacteria(inmillions)leftafterxdaysif5daysagotherewereapproximately64.3millionandthedeathratefromadrugtreatmentis40%perday.

5. a.

4

16y

x

(2, 16)

b.y-interceptaty 5 12;thedomain-restrictedfunctionhasnox-intercepts(theunrestrictedfunctionhasinterceptsatx 5 22andx 56);noasymptotes

c.7y 16

6. a.

4

20

40 (3, 31)

y

x

b.y-interceptaty540;nox-intercepts;noasymptotes

c.31y56

7. a.

4

y

x

(4.36, 20.75)

20

20

b.y-interceptaty512;x-interceptsatx521,x52,andx56;noasymptotes

c.220.7453y40

8. a.

40

y

x3 3

2 Problem Set 1-2 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press

PC3_SM_Ch01.indd 2 6/23/11 1:40:20 PM

b.y-interceptaty 516;thedomain-restrictedfunctionhasx-interceptsatx522,x51,andx52(theunrestrictedfunctionhasanadditionalinterceptatx524);noasymptotes

c.220y70

9. a.

4 8

4

8

12y

x

b.y-interceptaty50;x-interceptatx50;noasymptotes

c.0y12

10. a.

4 8

4

8

y

x

b.y-interceptaty50;boththedomain-restrictedfunctionandtheunrestrictedfunctionhaveanx-interceptatx50;noasymptotes

c.0y8.1

11. a.

4 8

3

y

x

b.y-interceptaty54;x-interceptatx555__ 7;noasymptotes

c.23y6.1

12. a.

10

20y

x4 4

b.y-interceptaty56;x-interceptatx 522;noasymptotes

c.29y21

13. a.

4

8

y

x4 4

b.y-interceptaty53;nox-intercepts;asymptotey50(thex-axis)

c.0.8079y11.1387

14. a.

50

100

y

x4 4

b.y-interceptaty 520;nox-intercepts;asymptotey 50(thex-axis)

c.3.3614 y 118.9980

15. a.

4

10

20

y

x

b.Noy-intercept;nox-intercept;asymptotesx50(they-axis)andy50(thex-axis)

c.y 0

16. a.

4 8

20

40

y

x

b.y-interceptaty 50;x-interceptatx 50;noasymptotes

c.y 0

17. a.

2 4

y

x

4

4

b.y-interceptaty 51__ 2;x-interceptatx52;asymptotesx521,x54,andy 50(thex-axis)

c.Range:allrealnumbers

18. a.

2 4 6

y

x

8

4

4


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b.y-interceptaty 52__ 3;x-interceptatx 51__

35 20.7320or2.7320;asymptotex 53(andslantasymptotey 5x 11)

c.Range:allrealnumbers

19. Exponential

20. Linear

21. Linear

22. Exponential

23. Quadratic(polynomial)

24. Cubic(polynomial)

25. Power

26. Inversevariation

27. Rational

28. Directvariation

29. a. 30. a.

Weight

Length Time

Temp

b.Power(cubic) b. Exponential

31. a. 32. a.

Cost

sq ft s

ft

b.Linear b. Quadratic

33. Function;nox-valuehasmorethanonecorrespondingy-value.

34. Notafunction;somex-valuesonthelefthavetwocorrespondingy-values.

35. Notafunction;thereisatleastonex-valuewithmorethanonecorrespondingy-value.

36. Function;nox-valuehasmorethanonecorrespondingy-value.

37. Notafunction;thereisatleastonex-valuewithmorethanonecorrespondingy-value.

38. Notafunction;thex-valueinthemiddlehasinfinitelymanycorrespondingy-values.

39. a.Averticallinethroughagivenx-valuecrossesthegraphatthey-valuesthatcorrespondtothatx-value.So,ifaverticallinecrossesthegraphmorethanonce,itmeansthatthatx-valuehasmorethanoney-value.

b. (Sketchnotshown.)InProblem33,anyverticallinecrossesthegraphatmostonce,butinProblem35,anyverticallinebetweenthetwoendpointscrossesthegraphtwice.

40. Itisallrightinafunctionfordifferentxstoproducethesamey,butarelationisnotafunctionifthesamexproducesdifferentys.

41. x 2,thatis,thenumber(orthevariablerepresentingit)thatisbeingsubstitutedintof.

42. Studentresearchproblem

Problem Set 1-3Q1. Quadratic Q2.y 5a x b , a 0,b 0

Q3. y 5a b x , a 0, Q4. x 21x 256 b 0,b 1

Q5. 9x 2230x 125 Q6.

y

x

Q7. Q8.

4

6

y

x

Q9. 900 Q10.D

1. a. g (x )52_______

92x 2

b.

y

x5 5

5

c.y-dilationby2(outsidetransformation)

2. a. g (x )5231______

9x 2

b.

y

x5 5

5

c.y-translationby23(outsidetransformation)

3. a. g (x )5____________

92(x 24)2

b.

y

x5

5

5

c.x-translationby4(insidetransformation)


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4. a. g (x )5________

92(x __ 3)2

b.

y

x5 5

5

c.x-dilationby3(insidetransformation)

5. a. g (x )511________

92(x __ 2)2

b.

y

x5

5

5

c.x-dilationby2(insidetransformation),y-translationby1(outsidetransformation)

6. a.g (x )51__ 2

____________ 92(x 13)2

b.

y

x5

5

5

c.x-translationby23(insidetransformation),y-dilationby1__ 2(outsidetransformation)

7. a. y-translationby7

b.g (x )571f (x )

8. a. x-translationby25

b.g (x )5f (x 15)

9. a. x-dilationby3

b.g (x )5f ( x __ 3)10. a. y-dilationby4

b.g (x )54f (x )

11. a. x-translationby6,y-dilationby3

b.g (x )53f (x 26)

12. a. x-dilationby3,y-translationby24

b.g (x )5241f (x __ 3)13. No.Thedomainoff (x ) isx 1,butthedomainofthegraph

is23x 1.Thatrestrictionmustbeaddedtothedefinitionoff (x ) .

14. No.Thedomainoff (x ) isx 1,butthedomainofthegraphis23x 1Thatrestrictionmustbeaddedtothedefinitionoff (x ) .

15. a.

10 4 4 104

2 x

y

b.x-translationby26

16. a.

4

2

y

x

b.x-dilationby2

17. a.

42

y

x

b.y-dilationby5

18. a.

42

y

x

b.y-translationby4

19. a.

4 4

2 x

y

b.y-dilationby5,x-translationby26


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20. a.

42

y

x

b.x-dilationby2,y-translationby4

21. Answerswillvary.

Problem Set 1-4Q1. y-dilationby3 Q2.y-translationby5

Q3. f (x 1 4) Q4.f (5x )

Q5. xisthebase,nottheexponent.

Q6. f (x ) 5a x 21bx 1c, a 0 Q7. x 55

Q8.

x

f(x)

1

1

Q9. 120 Q10.C

1. a. 517(4)533cm; 517(7)554cm

b. 33253421.1943cm2; 54259160.8841cm2

c.Theareadependsontheradius,whichinturndependsonthetime.Areaistheoutsidefunctionandradiusistheinsidefunction.

d.r(t)5517t;a (r(t))5 (r(t))2;a(r (t))5 (517t)2;a(4)5 (33)25108953421.1943cm2;a(7)5 (54)252916 59160.8841cm3

2. a. A (0)59(1.1)05 9mm2;A (5)59(1.1)5514.4945mm2;A (10)59(1.1)10523.3436mm2;

b. (R (0))259mm2 R (0)5___

9__ 51.6925mm;

(R (5))2514.4945mm2

R (5)5 __________

14.4945 __________ 5 2.1479mm;

(R (10))2523.3436mm2

R (10)5 __________

23.3436__________ 5 2.7258mm

c.Theradiusdependsonthearea(essentiallythenumberofbacteria),whichinturndependsonthetime.Radiusistheoutsidefunctionandareaistheinsidefunction.

d.a 5r 2r 25a __ r 5 6__

a __

Onlythepositivevaluemakessenseinthecontext,so

R (A (t ))5____

A (t )____ .

A (t )59(1.1 )t ,soR (A (t ))5______

9(1.1)t ______ .

R (A (5 ))5_______

9(1.1)5 _______ 52.1479mm

3. a.Answerswillvary.Notethatshoesizeisadiscretegraph,becauseshoesizescomeonlyinhalfunits.

Sampleanswer:

7 8 9

5

10

x (in.)

S(x) (size)

6 10 11

10

10

5

20 30 40 50 60 70 80x (yr)

L(x) (in.)

b. InS(x),xrepresentsfootlength(ininches,forthegraphabove).InL(x),xrepresentsage(inyears,above).ThecompositefunctionS (L(x))givesshoesizeasafunctionofage(xrepresentsage).L (S(x))wouldbemeaninglesswiththegivenfunctionsLandS.BecausexissubstitutedintoS,xmustrepresentfootlength.Sthengivesshoesize.ButthisissubstitutedintoL,whichexpectstohaveanage,notashoesize,substitutedintoit.(Ifwehadtwocompletely differentfunctions,SgivingshoesizeasafunctionofageandLgivingfootlengthasafunctionofshoesize,thenL (S(x))wouldgiveusfootlengthasafunctionofage.)

c.Answerswillvarybutshouldbethecompositeofthegraphsinparta.Again,shoesizeisadiscretegraph.

Sampleanswer:

10 20 30 40 50 60 70 80x (yr)

S(L(x))

5

10

90


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4. a.ThegraphofT(x)shouldbesimilartotheonebelow.ThegraphsofS(x)mayvary.Asamplegraphisshown.

20 40 60 80

5

10

x (mi/h)

T(x) (min)

50 100

50

x (cars per mile)

S(x) (mi/h)

b. InT(x),xrepresentsmilesperhour.InS(x),xrepresentsthenumberofcarspermile.ThecompositefunctionT (S(x))givesthetimetotravel1miasafunctionofthenumberofcarspermile(xrepresentscarspermile).S (T(x))wouldbemeaninglesswiththegivenfunctionsSandT,becauseT(x)representstimeinminutesandtheinputtoSmustbenumberofcarspermile.

c.Answerswillvarybutshouldbethecompositeofthetwofunctionsinparta.Sampleanswer:

20 40 60 80 100

5

10

x (cars per mile)

T(S(x)) (min)

5. a.h (3)55

1 2 3 4 5 6 71234567

x

h(x)

b.p (h (3))5p (5)53.5

1 2 3 4 5 6 7

1234567

x

p(x)

c.p ( h (2))5p (3)54.5;p ( h (5))5 p (4)54

d.h ( p (2))5h (5)54,whichisdifferentfromp ( h (2))54.5,foundinpartc.

1 2 3 4 5 6 71234567

x

p(x)

1 2 3 4 5 6 71234567

x

h(x)

e.h ( p (0))5h (6),whichisundefined,because6isnotinthedomainofh.

6. a.g (4)48

1 2 3 4 5 610

203040

5060

x

g(x)

b.f ( g (4))f (48)51

20 40

x 4860

50100150200250300

x

f (x)

c.f ( g (3))f (39)75;f ( g (2))f (28)150 d.f ( g (6))isundefinedbecause6isnotinthedomain

ofg.


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e.f ( g (5))f (55),whichisundefinedbecause55isnotinthedomainoff .

1 2 3 4 5 610

203040

5060

x

g(x)

20 40 6050100150200250300

x

f (x)

7. a. g (1)52;f ( g (1))5f (2)55 b.g (2)53;f (g (2))5f (3)54 c.g (3)57;f ( g (3))5f (7),whichisundefined. d.f (4)52;g ( f (4))5g (2)53 e.g ( f (3))5g (4)55 f.f ( f (5))5f (1)53 g.g ( g (3))5g (7),whichisundefined. h.f ( f ( f (1)))5f ( f (3))5 f (4)5 2

8. a.v (2)56;u ( v (2))5u (6)52

b.v (6)54;u ( v (6))5u (4)58

c.v (4)55;u ( v (4))5u (5),whichisundefinedbecause5isnotinthedomainofu.

d.u (4)58;v (u (4))5v (8)52

e.v ( u (10))5v (6)5 4

f.v ( v (10))5v (8)5 2

g.u ( u (6))5u (2)5 3

h.v ( v ( v (8)))5 v ( v (2))5v (6)5 4

9. a. x g(x) f(g(x)) 1 3 none

2 4 5

3 5 4

4 6 3

5 7 2

b.Thedomainoff g appearstobe2 x 5.Domainoff g: 4g(x )8 4x 1 2 82x 6; theintersectionofthiswiththedomainofg,1 x 5,gives2 x 5.

c.6isnotinthedomainofg,sog(x )isundefined.g(1)5 (1)1 25 3, but3isnotinthedomainoff.

d.

x f(x) f(g(x)) 4 5 7

5 4 6

6 3 5

7 2 4

8 1 3

Thedomainofg fappearstobe4x 8.Domainofg f:1 f(x) 5 192 x 5 282x 24 4x 8;theintersectionofthiswiththedomainoff,coincidentallyalso4x 8,gives4x 8.

e.

4 8

4

y

x

g

f

g f

f g

Thedomainsofthecompositefunctionsmatchtheresultsinpartsbandd.

f. f (f (5))5f (925)5f (4)59245 5;g (5)5 5125 7,and7isnotinthedomainofg.

10. a. x g(x) f(g(x)) 0 5 11

1 4 12

2 3 11

3 2 8

4 1 3

5 0 none

6 21 none

7 22 none

b.1g(x)61 52 x 6 24 2x 1 21 x 4; theintersectionofthiswiththedomainofg,0 x 7,is0 x 4,whichagreeswiththetable.

c.f(g(3))5 f (523)5 2(2)21 8(2) 2 45 8; g (f (3))5 g (2(3)21 8(3)2 4)5 g (11),but11isnotinthe

domainofg,sog (f (3))isundefined. d.

2 4

5

10f

g

f g

y

x

e.f ( g (x ))5 f (52x )5 2(52x )21 8(52x )2 45 2 x 21 2x 1 11,withthedomain0 x 4foundinpartb.Thegraphcoincideswiththegraphinpartd.

11. a.f ( g (3))5 f (__

3)5 (__

3)25 3

f (g (7))5 f (__

7)5 (__

7)25 7

g ( f (5))5 g ( (5)2))5 g (25)5 ___

255 5

g ( f (8))5 g ( (8)2))5 g (64)5 ___

645 8

Conjecture:Forallvaluesofx, f ( g (x ))5 g ( f (x ))5 x .


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b. 29isnotinthedomainofg,sog(29)isundefined,sof (g (29))isundefined.

g (f (29))5 g ((29)2)5 g(81)5 ___

815 9 29.No.

c.

2

4

y

x

f

g

f g

2 2

f ( g (x ))5f ( __

x ) 5(__

x )2 5 x, butgisdefinedonlyfornonnegativex,sof g isdefinedonlyfornonnegativex.

d.

2

4

y

xg

f

g fg f

2 2

e.g (f (x ))5 g (x 2)5 ___

x 25

x ifx 02 xifx 0

5 x

12. a.Translation3unitstotheright

b.Horizontaldilationbyafactorof2

c.

3f

f gf h

3 3

y

x

Yes

13. Ifthedottedgraphisf (x ),1x 5,thenthesolidgraphisg (x )5 f (2x ),25x 21.Intermsofcompositionoffunctions,thesolidgraphisg (x )5 f (h (x )),whereh (x )5 2x .

14. Ifthedottedgraphisf (x ),1x 5,thenthesolidgraphisg (x )5 f (x ), 1x 5.Intermsofcompositionoffunctions,thesolidgraphisg (x )5 h (f (x )),whereh (x )5 x .

15. a.f (g (6))5f ( 1.5(6)13)5f (12)52__ 3(12)2256;

f (g (215))5f (1.5(215)13)5f (219.5)

5 2__ 3(219.5)225215;

g (f (10))5g (2__ 3(10)22)5g (42__ 3)5 1.5(42__ 3)1 3510;g (f (28))5g ( 2__ 3(28)2 2)5g ( 271__ 3)5 1.5(271__ 3)1 3528. f (g (x ))5 g (f (x ))5 x

b.

f

gf g g f

4

4

4

y

x

f ( g (x ))andg (f (x ))coincidewitheachother,andwiththeliney 5x .f (x )andg (x )areeachothersreflectionsacrossthatline.

c.f (g (x ))5f (1.5x 13)52__ 3(1.5x 13)22

52__ 33__ 2x 1

2__ 33225x 12225x ;g (f (x ))5g ( 2__ 3x 22)51.5(2__ 3x 22)1353__ 2

2__ 3x 13__ 2(22)135x 23135x

d.Findj (x )suchthath (j (x ))5 x

5j (x )2 75x 5j (x )5x 17 j (x )5 x 1 7______ 5 5

1__ 5x 1 7__ 5.

Check:h (j (x ))5 h ( 1__ 5x 17__ 5)

55(1__ 5x 17__ 5)2 75 x 172 75x ,andj (h (x ))5 j (5x 27)51__ 5(5x 27)1

7__ 55x 27__ 51

7__ 55 x .

Problem Set 1-5Q1. Inside Q2.Outside

Q3. (m d )(x ) Q4.8

Q5. 5 Q6.4

Q7. 2 Q8.y5x

Q9. 1 Q10.g (x )52x13

1. a.f (5)524psi;f (10)516psi;f (15)510.7psi

b.Theairleaksoutofthetireastimepasses,sothepressureisconstantlygettinglower.Thus,fisadecreasingfunctionandhenceisinvertible.f 21(24)55min,whichanswersthequestionAtwhattimewasthepressure24psi?f 21(16)510min,whichanswersthequestionAtwhattimewasthepressure16psi?

c.Somewherebetweenx 5 25andx 5 30min,alltheairgoesoutofthetire,andthepressureremains0.Soitisnotpossibletogiveauniquetimecorrespondingtoapressureof0psi;f 21(0)cannotbedefined.

d.Thegraphoftheinverserelationisdotted.Thetwographsarereflectionsofeachotherovertheliney 5x .(Theycoincidentallyhappentobeverycloseovermostoftheirlength.)

10 20 30 40

10

20

30

40

x

y

y f(x)y x


PC3_SM_Ch01.indd 9 6/23/11 1:40:41 PM

e.Asaninputforf ,x representstimeinminutes.Asaninputforf 21,itrepresentspressureinpsi.

2. a.c (40)55c/min;c (50)5 30c/min;c (60)555c/min

b.Anyone-to-onefunctionisinvertible.c 21(30)5 50F;c 21(80)570F;thesegivethetemperaturecorrespondingto30c/minand80 c/min.Bycontrast,c (30)andc (80)givethenumberofchirps/mincorrespondingto30Fand80F(0c/minand105c/min).

c.Thecricketdoesnotbeginchirpinguntilthetemperatureisatleast30F.Atleastfor20x 30,thenumberofchirps/minremains0,soc 21(0)cannotbedefined.

d.

50 100

50

x

y

y c(x) y x

Thegraphsarereflectionsofeachotheracrosstheliney 5x .

e.Astheinputtoc ,x representstemperatureinF.Astheinputtoc 21,itrepresentsnumberofchirps/min.

3.

10 20

10

20

x

y

Throughoutmostofitsdomain,theinverserelationhastwoy-valuesforeveryx-value.

4. Noy-valuecomesfrommorethanonex-value.Also,nohorizontallinepassesthroughmorethanonepointofthefunction.

5. Function

y

x

y x

6. Notafunction

y

x

y x

7. Notafunction

y

x

y x

8. Notafunction

y

x

y x

9. a.

5

y10

5

5

5x

b. Notafunction

c. Graphergraphagreeswithgraphonpaper.


PC3_SM_Ch01.indd 10 6/23/11 1:40:44 PM

10. a.

5

10

y

x5 5

b. Notafunction


11. a.

5

y

x

10

10 5

5

5

b. Notafunction


12. a.

5

y10

5 510x

5

b. Function


13. a.

5

y

x

10

10

5

5

b. Function


14. a.

5

10y

x105

5

b. Notafunction


15. a. Thex -coordinatesareequalifandonlyif

t 1151.5t 12

t 522s

Whent 522s,Particle1andParticle2arebothatx 521mandy 53m.Pathsintersectsimultaneouslyatpoint(21,3)whent 522s.Pathsintersectatpoint(2,6)butnotsimultaneously.

b. Graphergraphconfirmsthatthepathsintersectsimultaneouslyonlyatpoint(21,3)whent 522s.

16. a. Equatingy -valuesgives

1015t 510(t 22)

t 56h

Substituting6fort gives

Freighter:x 590210(6)530mi

Cutter:x 58(622)532mi,whichisnotequaltothefreightersx -value.

b. Theshipsdonotarriveattheintersectionpointatthesametimebecausethetwox -valuesarenotequalwhenthetwoy -valuesareequal.

Tofindtheintersectionpoint,eliminatetheparametertfrombothpairsofparametricequations.

Freighter:t 5920.1x ,whichgivesy 55520.5x.

Cutter:t 5210.125x ,whichgivesy 51.25x.

Set1.25x 55520.5x ,whichgivesx 531.4285mi(agreeingwiththegivengraph).

Freighterarrivesatx 531.4285miwhent 55.8571h.

Cutterarrivesatx 531.4285miwhent 55.9285h.

Freighterarrivesattheintersectionpoint0.0714hour,orabout4minutesbeforecutter.


PC3_SM_Ch01.indd 11 6/23/11 1:40:46 PM

17. Function

4

4

4 4

y

x

8

8

8

18. Function

4 4

4

4

y

x

19. Notafunction

4 4

4

4

y

x

20. Notafunction

4 4

4

4

y

x

21. Function

4 4

4

4

y

x

22. Function

4 4

4

4

y

x

23. Function

4 4

4

4

y

x

24. Function

y

x

4

4

4 8

25. Function

4 4

4

4

y

x

26. Function

f

f

f 1 f 14 4

4

4

y

x

27. Function

4

4

y

x


PC3_SM_Ch01.indd 12 6/23/11 1:40:50 PM

28. Notafunction

5 10

46810

2

42

5

y

x

29. x 52y 2 62y 5x 16

y 5f 21(x )51__ 2x 13

y

xf1

f4

4

4

4

Theinverserelationisafunction.

30. x 520.4y 1420.4y 5x 24y 5 f 21(x )

5x 2 4______ 20.45 22.5x 1 10

8

x

y

f

f 14 8

Theinverserelationisafunction.

31. x 520.5y 222y 25x 1 2______ 20.5 y 56________

22x 2 4

5

5

5

5

y

x

f

Theinverserelationisnotafunction.

32. x 5 0.4y 213 y 25x 2 3______ 0.4

y 56__________

2.5x 27.5

5

5

5

5

y

x

f

Theinverserelationisnotafunction.

33. f (f (x ))5 1___ f (x )51____

(1/x )5 x , x 0

34. f (f (x ))5 2f (x )5 2(2x )5 x forallx.

35. a.c (1000)5 900.Ifyoudrive1000miinamonth,yourmonthlycostis$900.

b. c 21(x )5 2.5x 2 1250.c 21(x )isafunctionbecausenoinputproducesmorethanoneoutput.c 21(758)52.5(758)212505645.Youwouldhaveamonthlycostof$758ifyoudrove645miinamonth.

c.

y

x

c(x)

c1(x)200

200 600 1000

400600800

1000

36. a.A (50)55.4288;A (100)5 8.6177;A (150)5 11.2924Deerthatweigh50,100,and150lbhavehidesofareasapproximately5.43,8.62,and11.29ft2,respectively.

b.False.A (100) 2A (50)

c.Interchangethevariablesiny 5A (x )50.4x 2/3:

x 5 0.4y 2/3 y 5( x ___ 0.4)3/2 5 (2.5x )1.5 A 21(x )5 (2.5x )1.5

d.

100 200

100

200

y

x

A1(x)

A(x)

y = x

Thetwocurvesarereflectionsofeachotheracrosstheliney 5x .

37. a.x 50.057y 2 y 25 x ______ 0.057

y 5d 21(x )5______

x ______ 0.057.Becausethedomainofdisx 0,

therangeofd 21isd 21(x )0.

b. d 21(200)5 59.234Thismeansthata200-ftskidmarkiscausedbyacarmovingataspeedofabout59mi/h.

c.

50

50d(x)

x

d.Becausethedomainofdnowcontainsnegativenumbers,therangeoftheinverserelationcontainsnegativenumbers.Now,becausetherangeoftheinverserelationcontainsnegativenumbers,

y 56

______ x ______ 0.057,whichisnotafunction.


PC3_SM_Ch01.indd 13 6/23/11 1:40:54 PM

38.

x

y

x

y

Invertible Notinvertible

Problem Set 1-6

Q1. 1__ 2x Q2.x 1 3

Q3.x 1 3______ 2 Q4.y 56

__ x

Q5. Therearetwoy-valuesforeverypositivex-value.

Q6. 3 Q7. Invertible

Q8. Afunctionforwhicheachy-valueintherangecorrespondstoonlyonex-value

Q9. Sampleanswer:5 Q10. Sampleanswer:25

1. a.

y

x5 5

5

5

b.

y

x5 5

5

5

c.

y

x5 5

5

5

d.

y

x5 5

5

5

2. a.

y

x5 5

5

5

b.

y

x5 5

5

5

c.

y

x5 5

5

5

d.

y

x5

5

5

5

3. a.

y

x6 4

50

50

b.

y

x6 4

50

50

c.

y

x6 4

50

50


PC3_SM_Ch01.indd 14 6/23/11 1:40:59 PM

d.

y

x6 4

50

50

4. a.

y

x5 5

5

5

b.

y

x5 5

5

5

c.

y

x5 5

5

5

d.

y

x5 5

5

5

5. Thegraphsmatch.

6. Thegraphsmatch.

7. a.

y

x5 5

5

5

Thistransformationreflectsallthepointsonthegraphbelowthex-axisacrossthex-axis.

b.

y

x5 5

5

5

Thistransformationreflectsf(x),forpositivevaluesofx, acrossthey-axis.

c.f (3) 50.5(32 2)2 24.55 24 54; f (23 ) 50.5(232 2)22 4.5 5 0.5(32 2)

224.55 24

23isnotinthedomainoff ,but2353isinthedomainoff .

d.

y

x5 5

5

5

y

x5 5

5

5

8. a.

40 80 120

y

x100

200

b.f (10)52140m;f (40)5 70m.Attimex 510,heis140metersbefore(behindorbelow)thegasstation.

c.140mand70m,respectively.Theanswersarepositivebecausedistanceisalwayspositive.

d.d (x )5 0.1x 2112x 2250

100

200

40 80 120

y

x

e.x 593.1662 s


PC3_SM_Ch01.indd 15 6/23/11 1:41:04 PM

9. a.

4 4

8

4

4

8

x

y

Thepolynomialfunctionf (x )isthesumofevenpowersofx .Anegativenumberraisedtoanevenpowerisequaltotheabsolutevalueofthatnumberraisedtothesamepower.So,for6x,thesamecorrespondingy-valueoccurs,andthereforef (x )5 f (2x ).

b.

4 4

8

4

4

8

x

y

Anegativenumberraisedtoanoddpowerisequaltothe

oppositeoftheabsolutevalueofthatnumberraisedtothesamepower.Becauseeachterming(x)isamonomialinxraisedtoanoddpower,g(2x)hasthesameeffectong(x)as2g(x).

c.Functionhisodd;functionjiseven.

d.

y

x5

5

5

5

Thefunctione(x)isneitheroddnoreven.e (2x ) 5 22x 2x ,and 22x 2 2 x

10.

y

x

Thefunctionfisanevenfunction.

f (2x )52x 5 x5 f (x )

11. a.Thegraphsmatch.

b.g (x )53x 2 4 ________ x 2 4 1 5;g (x )53f (x 24)15

c.f (x )5 (x 2 3)222x 2 5 ________ x 25

Thegraphsmatch.

12. a.

3

3

y

x

f (2.9)52,f (3)5 3,f (3.1)5 3

b.

1 2 3 4 5

40

80

120

Weight (oz)

Price (cents)

c.Dilatedbyafactorof23;translatedup37cents;

y 5

0,x 5 0

2232x 11137,x 0

Thegraphsmatch.

d.2232x 111 37 313 2232x 11 276 2x 11 212 2x 11 212 2x 213 x 13

So0 x 13.

e.Answerswillvary.

13. a.a 40005150a 5150_____ 400050.0375;

b _______ 40002

5 150 b 5 15040002

5 2,400,000,000

b. f 1 (x )50.0375x/(0x andx 4000);

f 2 (x )52,400,000,000/x 2/ (x 4000)

4000 8000

50

100

150y

x

c.y (3000)50.0375(3000)5 112.5lb;

y (5000)5

2,400,000,000_____________

50002 5 96lb

d. f 1 (x )5 0.0375x 550 x 5 50_______

0.03755 1.333.

__ 3mi;

f 2 (x )52,400,000,000

_____________ x 2 550 x 5

______________

2,400,000,000

_____________ 50

5 6928.2032 mi



PC3_SM_Ch01.indd 16 6/23/11 1:41:07 PM

Problem Set 1-7 1. Journalentrieswillvary.

Problem Set 1-8

Review Problems

R1. a. 17.15psi;5.4min

b. x y

0 35

1 24.5

2 17.15

3 12.005

4 8.4035

c.Thegraphintersectstheliney 55psiatapproximatelyx 55.5min.

Domain:0 x 5.5;range5 y 35. d.Asymptote

e.

Stress

Time

R2. a.Linear

b.Polynomial(cubic)

c.Exponential

d.Power

e.Rational

f.Answerswillvary;e.g.,numberofitemsmanufacturedandtotalmanufacturingcost.

g.13f (x )37

h.

y

x

Aquadratic(withanegativex 2-coefficient)fitsthispattern.

i.1-8b:exponential;1-8c:polynomial(probablyquadratic);1-8d:power

j.Figure1-8epassestheverticallinetest:noverticallineintersectsthegraphmorethanonce,sonox-valuecorrespondstomorethanoney-value.Figure1-8ffailstheverticallinetest:thereisatleastoneverticallinethatintersectsthegraphmorethanonce,somorethanoney-valuecorrespondstothesamex-value.

R3. a.Horizontaldilationbyafactorof3,verticaltranslationby25;

g (x )5____________

42(x __ 3)22 5

b.Horizontaltranslationby14,verticaldilationbyafactorof3

5

5

x

y

f

g

R4. a.h (t )5 3t 1 20

b.h (5)5 3(5)1 205 35in.

W (h (5))5 0.004(35)2.5 29lb c.

8

40

t

y

d.No;thegraphiscurved.

e.Answerswillvary.Possibleanswer:0 t 13

f.f (g (3))5 f (4)5 6;f (g (4))5 f (5)5 3;f (g (5))5 f (8),whichisundefined;f (g (6))5 f (3)5 2;g (f (6))5 g (5)5 8;

f (f (3))5f (2),whichisundefined; g ( g (3))5g (4)5 5 g.

5

5

y

x

f

g

f g

h.f (g (4))5 f (2(4)2 3)5 f (5)5 (5)2 25 3 i.f ( g (3))5 f ( 2(3)2 3)5 f (3),whichisundefined,because

3isnotinthedomainoff.

j. 4g (x ) 8 4 2x 2 3 8 7__ 2 x 11__ 2.The

intersectionofthiswiththedomainofg,2 x 6,is7__ 2 x

11__ 2,whichagreeswiththegraph.

R5. a.Theinversedoesnotpasstheverticallinetest.

y

x

5

5

5 5


PC3_SM_Ch01.indd 17 6/23/11 1:41:09 PM

b.

5 5

5

5

x

y

Thegraphsareeachothersreflectionsacrosstheline.Thedomainoffcorrespondstotherangeoftheinverserelation.Therangeoffcorrespondstothedomainoftheinverserelation.

c.x 5 y 211 y 56______

x 2 1.The6revealsthattherearetwodifferenty-valuesforsomex-values.

d.

55

5

5

y

x

Itpassesthehorizontallinetest;asymptotes.

e.

10

5

y

x

10

5

Graphergraphagreeswithgraphonpaper.Notafunctionbecauseitfailstheverticallinetest;everyx inthedomainhasmultiplevaluesofy .

f.

4

3

2

1

4321

Thecurveisinvertiblebecauseitisincreasing.Astheinputtov,xrepresentsradiusinmeters.Astheinputtov 21,itrepresentsvolumeincubicmeters.Ifx 0isaparticularinputtov,then(x 0,v (x 0 ))isapointonthegraphofv (x ).Pluggingtheoutput,v (x 0),intov

21givesthepoint(v (x 0),v 21(v (x 0)))onthegraphofv 21(x ).Butthegraphofv 21(x )isjustthegraphofv (x )withallthex-andy-valuesexchanged,sothispointisactually(v (x 0),x 0).Thus,v 21(v (x 0))5x 0.

g.Sincenoycorrespondstomorethanonexintheoriginalfunction,noxcorrespondstomorethanoneyintheinverserelation,sotheinverserelationisafunction.

Samplegraph:

y

x

R6. a.

y f(x)

y

x5 5

5

5

y f(x)

y

x5 5

5

5

y | f (x)|

y

x5 5

5

5

y f(|x|)

y

x5 5

5

5

b.ThegraphagreeswithFigure1-8k;eachofthegraphsagreeswiththoseinparta.

c.Becausepowerfunctionswithoddpowerssatisfythepropertyf (2x )5 2f (x )andpowerfunctionswithevenpowerssatisfythepropertyf (2x )5 f (x )

d.

6

4 4

y

x

Discontinuity

R7. Answerswillvary.


PC3_SM_Ch01.indd 18 6/23/11 1:41:13 PM


T7. Odd

T8. Neither

T9. Horizontaldilationby2;g (x )5f ( x __ 2)

T10. Horizontaltranslationby21,verticaltranslationby15;g (x )5 f (x 11)1 5

T11. Horizontaltranslationby16,verticaldilationby2;g (x )5 2f (x 26)

T12. Domain:22 x 7;range:4 y 9

T13. Verticaldilationby1__ 2

5

5

y

x

T14. Horizontaldilationby3__ 2

5

5

y

x

T15. Horizontaltranslationby23,verticaltranslationby24

5

5

y

x

T16. Reflectionthroughtheliney 5x

5

5

y

x

T17. Thegraphfailstheverticallinetest.(Thepre-imagegraphfailsthehorizontallinetestitisnotone-to-one.)

Concept Problems

C1. Horizontaldilationby3(widthfrom4unitsto12units),verticaldilationby2(heightfrom4unitsto8units),horizontaltranslationby13,verticaltranslationby25;

g (x )5 2f (1__ 3(x 23))2 55 2(x 2 3______ 3 )22 5C2. a.Answerswillvary.Thefunctionrepeatsitself

periodically.

b.About6.3,or2

c.Odd.Itisitsownreflectionthroughtheorigin,sof (2x )5 2f (x ).

d.

1010

5

5

y

x

y 5 5sin(x )

e.Horizontaltranslation12,verticaltranslation13[(0,0)movesto(2,3)];y 5 sin(x 2 2)1 3

f.Horizontaldilationby2

y

x

f g

1

Chapter Test

T1. Exponential

T2. Linear

T3. Polynomial(quadratic)

T4. Power

T5. AllexceptT3.Functionsthatarenotone-to-onearenotinvertible;thatis,theirinversesarenotfunctions.

T6. Answerswillvary.

Time

Temperature

or

Time

Temperature

PC3_SM_Ch01.indd 19 6/23/11 1:41:17 PM


T24. x 5 3.2y 0.52y 5( x ___ 3.2) 1____ 0.52

Ifyouknowthepercentagelossandwanttofindthenumberofwildoatplantspersquaremeter

T25. L 21(100)5(100____ 3.2)1____ 0.525749.3963

Ifthecroplossis100%(i.e.,thetotalcropislost),theremusthavebeenabout750wildoatplants(ormore)persquaremeter.

T26. Domain:0x 750;range:0y 100

T27.

500

500

x

y

L1(x)

L(x)

T28. Itpassestheverticallinetest.(Theoriginalfunctionpassesthehorizontallinetestitisone-to-one.)


T18. f (g (3))5f ( (3)22 4)5f (5)5__

5;

g (f (3))5 g (__

3)5 (__

3)22 4521; f ( g (1))5 f ( (1)22 4)5 f (23),whichisnotdefined,because

23isnotinthedomainoff.

T19. Horizontaltranslationby14,verticaltranslationby15,

andverticaldilationby3ofx ____ x ;y 5 3 x 2 4 ________ x 2 4 15

T20.

10

5

y

x

10

5

Graphergraphagreeswithgraphonpaper.Function,becauseitpassestheverticallinetest.

T21. L (x )variesproportionatelytothe0.52powerofx.Powerfunction.

T22. L (150)5 3.2(150)0.525 43.3228Ifthereare150wildoatplantspersquaremeterofland,thepercentagelosstothewheatcropwillbeabout43%.

T23. 60%ofthecropmeansa40%croploss.Solve405 3.2x 0.52toget

x 5(40___ 3.2)1____ 0.525 128.6596

About129plantspersquaremeter

PC3_SM_Ch01.indd 20 6/23/11 1:41:18 PM

Problem Set 2-1 1.

1 2 3 4 5

5

x

f(x)

Thehollowsectionisupward.Thebacteriaaregrowingfasterandfaster.

2.

1 2 3 4 5 6

5

10

15

x

g (x)

TheexponentialfunctiongraphinProblem1looksasthoughitisapproachingaverticalasymptote(althoughitisreallygrowingveryfastanddoesnthaveanasymptote),whilethepowerfunctiongraphappearsasthoughitisunbounded.False;foreachfootitincreasesinlength,theweightincreasesbythatamountcubedinpounds.

3.

10 20 30

20

40

60

x

q(x)

10

Thegraphisconcavedown.Thisgraphpossessesamaximum(highpoint)atx5131__ 3.

4.

3 6 9 12 15 18

40

80

x

h(x)

Becausethegraphisneitherconcaveupnorconcavedown,thecostperadditionalminuteisalwaysthesame.

Problem Set 2-2Q1. f (3)59 Q2. f (0)50

Q3. f (23)59 Q4.g(3)58

Q5. g(0)51 Q6.g(23)50.125

Q7. h(25)55 Q8.h(0)50

Q9. h(29)isundefined Q10.D

1. Inpowerfunctions,theexponentisconstantandtheindependentvariableisinthebase.Inexponentialfunctions,thebaseisconstantandtheindependentvariableisintheexponent.

2. Quadraticfunctionshaveeitheramaximumoraminimumpoint.Exponential,linear,andmanypowerfunctionsdonothavethese(exceptforcertainpowerfunctions,suchasy5x4/3andy5x3/2,thathaveaminimumpointattheorigin).

3. Answerswillvary.ThetermconcaveisfromtheLatincavus,meaninghollow.Theconcavesideofacurvedportionofagraphistheinsideofthatcurve.

4. Directvariationpowerfunctionshavetheformy5axnwithn0,soy50whenx50.Butinversevariationpowerfunctionsareundefinedatx50.

5. 1__ x5x21

6. Thisrestrictionexcludesstraightlinesfrombeingcalledquadratic.

7. (264)1/2isundefined,becauseitisthesquarerootofanegativenumber,but(264)1/3524,because(24)35264.Therestrictionallowsthefunctiontobedefinedforallvaluesofx.

8. Thegrapheronlyallowsyoutoenterequationsiny5form.Thesecondformshowsthehorizontaltranslationhandtheverticaltranslationk.

9. a. y55610.6(x220)50.6x144

b.Page44

c.0.6x144563x5312__ 3min5112__ 3minfromnow.

10. a.y21485a(x23)2 421485a(023)2 a52144_____ 9 5216y21485216(x23)2

b.y(5)5216(523)21148584ft

c.216(x23)2114850 216x2196x1450

x52(96)________________

(96)224(216)(4)__________________________

2(216)

56.0413...s,becauseonlythepositive answerapplies.

11. a.Linear

b. Increasingforallreal-numbervaluesofx,notconcave

c.Answerswillvary.

d.y52x27

e.Thegraphsmatch.

12. a.Linear

b.Decreasingforallreal-numbervaluesofx,notconcave

c.Answerswillvary.

d.y524x120

e.Thegraphsmatch.


Chapter 2 Properties of Elementary Functions

PC3_SM_Ch02.indd 21 6/23/11 1:45:01 PM

13. a.Quadratic

b.Decreasingforx2.25andincreasingforx2.25,concaveup

c.Answerswillvary.

d.y52x229x113

e.Thegraphsmatch.

14. a.Quadratic

b. Increasingforx4anddecreasingforx4,concavedown

c.Answerswillvary.

d.y5211___ 15x2127___ 5x1

311____ 15

e.Thegraphsmatch.

15. a.Exponential

b. Increasingforallreal-numbervaluesofx,concaveup

c.Answerswillvary.

d.y55(1.3)x e.Thegraphsmatch.

16. a.Exponential

b.Decreasingforallreal-numbervaluesofx,concaveup

c.Answerswillvary.

d.y596(0.5)x e.Thegraphsmatch.

17. a.Power

b. Increasingforx0,concavedown

c.Answerswillvary.

d.y55xlog21.6

e.Thegraphsmatch.

18. a.Power(inverse)

b.Decreasingforx0,concaveup

c.Answerswillvary.

d.y512x21

e.Thegraphsmatch.

19. a.Power

b. Increasingforx0,concaveup

c.Answerswillvary.

d.y53x3/2

e.Thegraphsmatch.

20. a.Linear(directvariation)

b. Increasingforallreal-numbervaluesofx,notconcave

c.Answerswillvary.

d.y50.8x

e.Thegraphsmatch.

21.

5 10 15

5

10

x

yz

Bothgraphsareconcaveupanddonotchangetheirconcavity,andeachbecomesinfiniteononesideoftheverticalaxis.Butthegraphproportionaltothesquareofxpassesthroughtheoriginandbecomesinfiniteonbothsidesoftheverticalaxis,whereastheexponentialfunctiondoesnotpassthroughtheoriginandbecomesinfiniteonlyonthepositivesideoftheverticalaxis.

22.

1 2 3

1

2

x

y

z

Bothgraphsareconcaveup,bothgraphsapproachzeroasxgrowslarge,andbothgraphsneverintersectthehorizontalaxis.Buttheexponentialfunctionintersectstheverticalaxis,whereastheinversegraphbecomesinfiniteanddoesnot.

23. Adirectvariationfunctioncanbewritteninthelinearformy5ax1bwithb50.Butyoucannotwritealinearfunctiony5ax1bwithb0asadirectvariationfunctiony5ax.

24. Youcanwriteapowerfunctionproportionaltothesquareofxinthequadraticformy5ax21bx1cwithb50andc50.Butyoucannotwriteaquadraticfunctiony5ax21bx1cwithb0orc0asapowerfunctiony5ax2.

25. 3e0.8x53e0.8x53b xb5e0.852.2255...

Thegraphsareequivalent.

Problem Set 2-3Q1. y5ax1b Q2. y5axb,a0,b0

Q3. y5axb,b0,b1,a0

Q4. y5ax21bx1c,a0 Q5. Power

Q6. Exponential Q7. Verticaldilationby4

Q8. B

Q9. Q10.y

x

y

x

1. Addaddproperty:linear

2. Multiplymultiplyproperty:power,inversevariation


PC3_SM_Ch02.indd 22 6/23/11 1:45:03 PM

3. Multiplymultiplyproperty:power;andconstant-second-differencesproperty:quadratic

4. Addmultiplyproperty:exponential

5. Addaddproperty:linear;multiplymultiplyproperty:power


7. Multiplymultiplyproperty:power,inversevariation


9. Addmultiplyproperty:exponential

10. Multiplymultiplyproperty:power

11. Constant-second-differencesproperty:quadratic

12. Constant-second-differencesproperty:quadratic

13. a. 65 b. 80 c. 1280

14. a. 360 b. 270 c. 1366.875

15. a. 70 b. 81 c. 72.9

16. a. 22600 b. 10 c. 0.1

17. f (8)513,f (11)519,f (14)525

18. f (6)55.6,f (12)544.8

19. f (10)5324,f (20)581

20. f (7)581,f (10)572.9,f (16)559.049

21. Multiplyyby4.

22. Multiplyyby16.

23. Divideyby2.

24. Divideyby4.

25. a.V(r)hastheformV5ar3wherea54__ 3p.Thevolleyballwouldhavevolume5400cm3.

b.Hisvolumewouldbe1000timesthatofanormalgorilla.400lb(10)35400,000lb5200tons

c.4000lb(100/20)35500,000lb d.200lb(1/10)350.2lb26. a.A5pr2.A(r)hastheformA5ar2wherea5p.

b.Thegrapefruitsrindwouldhavefourtimesasmuchareaasthatoftheorange.

c. 1___ 144.Theproportionoftheoriginallengthissquaredtofindtheproportionoftheoriginalarea.

d.2m2(10)25200m227. a. 42516timesmorewingarea.

b. 43564timesheavier.

c.Thefull-sizedplanehadfourtimesasmuchweightperunitofwingareaasthemodel.

28. a. A(2)5$1210,A(3)5$1331,A(4)5$1464.10

b.Tracethegraphofy51000(1.1)xtothepointwherey52000.Theinvestmentwilldoubleafterabout7years.

29. a. [H(3)2H(2)]2[H(2)2H(1)]5(1312121)2(121279)5102425232ft[H(4)2H(3)]2[H(3)2H(2)]5(1092131)2(1312121)52222105232ft[H(5)2H(4)]2[H(4)2H(3)]5(552109)2(1092131)52542225232ft

b.H(t)5216t2190t15;H(4)5109;H(5)555

c.H(2.3)5127.36ft;goingup;theheightseemstopeakataboutt53s.

d.H(t)5100t5290_____

2020______________ 232

51.4079...s(goingup)or4.2170s(comingdown).

e.Thevertexoftheparabolaisat

t52b___ 2a5290_______

2(216)52.8125s.

H(2.8125s)5131.5625ft

f.H(t)50t52902_____

8420______________ 232 55.6800...s

30. a. x f(x)

2 3(2)2512

4 3(4)2548

6 3(6)25108

8 3(8)25192

10 3(10)25300

b. f2(x)53x21100sinp__ 2x

2 4 6 8 10

100

200

300

x

y

f2(x)alsofitsthedata.

c. f3(x)53x2cospx

2 8 10

100

200

300

x

y

Manyfunctionscanfitasetofdiscretedatapoints.

31. [y(6)2y(5)]2[y(5)2y(4)]5(1127)2(725)52[y(7)2y(6)]2[y(6)2y(5)]5(17211)2(1127)52[y(8)2y(7)]2[y(7)2y(6)]5(27217)2(17211)54Ify(8)were25,thenaquadraticfunctionwouldfit.

32. a. x f(x)

1 20

2 14

3 8

4 8

5 20

6 50

b. D315[f (3)2f (2)]2[f (2)2f (1)]5(8214)2(14220)50D425[f (4)2f (3)]2[f (3)2f (2)]5(828)2(8214)56D535[f (5)2f (4)]2[f (4)2f (3)]5(2028)2(828)512D645[f (6)2f (5)]2[f (5)2f (4)]5(50220)2(2028)518D642D535D532D425D422D3156

c.Aquarticfunctionwillhaveconstantfourthdifferences.


PC3_SM_Ch02.indd 23 6/23/11 1:45:04 PM

33. Iff (x)5ax1b,thenf ( x2)5f ( x11c)1b5ax11ac1b5(ax11b)1ac5f ( x1)1ac.

34. Iff (x)5abx,thenf ( x2)5f ( cx1)5a(cx1)b5

a(cbx1b)5cbax1

b5cbf ( x1).35. Iff (x)5axb,then

f ( x2)5f (c1x1)5abc1x15a(bcbx1)5bcabx1

5bcf ( x1).36. f (x)5ax21bx1c;

f (x1d)5a(x1d)21b(x1d)1c5ax212adx1ad21bx1bd1c;f (x12d)5a(x12d)21b(x12d)1c5ax214adx14ad21bx12bd1c;f (x13d)5a(x13d)21b(x13d)1c5ax216adx19ad21bx13bd1c;

Firstdifferences: f (x1d)2f (x)

5(ax212adx1ad21bx1bd1c)2(ax21bx1c)52adx1ad21bd;f (x12d)2f (x1d)5(ax214adx14ad21bx12bd1c) 2(ax212adx1ad21bx1bd1c)52adx13ad21bd;f (x13d)2f (x12d)

5(ax216adx19ad21bx13bd1c)2 (ax214adx14ad21bx12bd1c)52adx15ad21bd

Seconddifferences:[f (x12d)2f (x1d)]2[f (x1d)2f (x)]5(2adx13ad21bd)2(2adx1ad21bd)52ad2;[f (x13d)2f (x12d)]2[f (x12d)2f (x1d)]5(2adx15ad21bd)2(2adx13ad21bd)52ad2

Problem Set 2-4Q1. Base Q2. Exponent

Q3. Exponentialexpression

Q4. x12

Q5. x2251__ x2 Q6. x35

Q7. Distribute Q8. 1___ 52

Q9. __

9 Q10.B

1. 1020.1549...50.7 2. 100.9030...58

3. 10a5b 4. a5logb

5. x51.574;101.574537.4973,log37.497351.574

6. x52.803;102.8035635.3309,log635.330952.803

7. x520.981;1020.98150.1044,log0.1044520.981

8. x523.58;1023.5850.0002630,log0.0002630523.58

9. x5log5751.7558;101.7558557

10. x5log35952.5550;102.5550...5359

11. x5log0.85520.0705;1020.070550.85

12. x5log0.0321521.4934;1021.493450.0321

13. 3.0277;103.027751066

14. 3.3012;103.301252001

15. 21.2247;1021.224750.0596

16. 20.5030;1020.503050.314

17. 0.001995;log0.001995522.7

18. 3162.2776;log3162.277653.5

19. 1.584831015;log(1.584831015)515.2

20. 102450.0001;log0.0001524

21. log(54)5log2051.301050.698910.60205log51log4;logxy5logx1logy;bcbd5bc1d

22. log(304)5log12052.079151.477110.60205log301log4;logxy5logx1logy;bcbd5bc1d

23. log(3547)5log550.698951.544020.8450

5log352log7;logx__ y5logx2logy;bc___ bd5bc2d

24. log(9646)5log1651.204151.982220.7781

5log962log6;logx__ y5logx2logy;bc___ bd5bc2d

25. log(25)5log3251.505155(0.3010)55log2;logbx;xlogb;bcd5bcd

26. log(43)5log6451.806153(0.6020)53log4;logbx5xlogb;bcd5bcd

27. log0.21520.6777520.52281(20.1549)5log0.31log0.7;0.2151020.677751020.52281(20.1549)51020.52281020.154950.30.7

28. log5651.748150.845010.90305log71log8;565101.74815100.845010.90305100.8450100.9030578

29. log650.778151.477120.69895log302log5;65100.77815101.477120.69895101.47714100.698953045

30. log1__ 4520.602050.301020.90305log22log8;0.2551020.60205100.301020.9030

5100.3010_________

100.903052__ 8

31. log3251.505155(0.3010)55log2;325101.5051510

5(0.3010)5(100.3010)5525

32. log12552.096953(0.6989)53log5;1255102.09695103(0.6989)5(100.6989)3553

33. log1__ 7520.845052log7;1__ 7510

20.84505 1_________ 100.8450

51__ 7

34. log0.00152352log1000;0.001510235 1____ 103

5 1_____ 1000

35. 73521 36. 5854037. 4841254 38. 442051__ 5

39. (845)35556 40. (2000440)4252541. 275128 42. 355243

43. 3,because125553 44. 6,because64526

45. Letc5logx,sox510c.Thenxn5(10c)n510cn,sologxn5cn5nc5nlogx.

46. Letc5logxandd5logy,sox510candy510d.Then

x__ y510c____ 10d

510c2d,sologx__ y5c2d5logx2logy.


PC3_SM_Ch02.indd 24 6/23/11 1:45:05 PM

47. a. 27 3356

9,612 343

415,316 3592

244,683,072

b.1018

c.8.3886;108.3886244,700,000,whichagrees(tofoursignificantdigits)withtheanswerfromparta.

Problem Set 2-5 Q1. 58540Q2. 364459

Q3. 5

Q4. 3

Q5. 131.5546.8721

Q6. Logos,arithmos

Q7. Exponent

Q8. 8,because48586Q9. 40,because0.5520440

Q10. E

1. logbx5yifandonlyifby5xforx0,b0,b1

2. logax5logbx______ logba

forx0,a0,a1,b0,b1

3. 7c5p

4. vx56

5. logk955

6. logm135d

7. log29______

log751.7304;check:71.7304529

8. log352_______

log8 52.8198;check:82.81985352

9. 6,because365729;alsolog729_______

log3 56

10. 1__ 5,because321/552;also

log2______

log3251__ 5

11. 5,because25532;alsolog32______

log255

12. 3,because535125;alsolog125_______

log5 53

13. log0.3_______

log6 520.6719;check:620.671950.3

14. log0.777_________

log15 520.0931;check:1520.093150.777

15. 8755616. 1020520017. 5

18. 9,because92581

19. 4,because364459

20. 14,because144752

21. 1__ 2,because__

x5x1/2

22. 1__ 5,because5

__ x5x1/5

23. 0,becausee051

24. 1,becausee15e

25. 1,because101510

26. 0,because10051

27. log107

28. ln0.07

29. 3

30. 0.5

31. log(orlog10)

32. ln

33. 3,becausek35k3

34. logk0.45logk(245)5logk22logk55x2y

35. log(3x17)503x17510051 3x 5 26 x522;check:log(3(22)17)5log150

36. 2log(x23)1155 log(x23)52 x2351025100 x5103;check:2log(10323)1152log100115221155

37. log2(x13)1log2(x24)53 log2[(x13)(x24)]53 (x13)(x24)52358 x22x21258 x22x22050 (x25)(x14)50 x155,x2524 Check:

1.x1:log281log21531053 2.x2:log2(21)1log2(28)isundefined.

38. log2(2x21)2log2(x12)521

log22x21_______ x12521

2x21_______ x12522151__ 2

2(2x21)5x12

3x54x54__ 3

Check:

log25__ 32log2

10___ 3

5log25__ 3___ 10__ 32log2

1__ 2521

39. ln(x29)458 4 ln(x29)58 ln(x29)52 x295e2 x591e2516.3890 Check:ln(e2)45lne858

40. ln(x12)1ln(x22)50 ln[(x12)(x22)]50 (x12)(x22)5e051 x22451 x255 x5

__ 552.2360

Check: 1.ln

__ 5121ln

__ 5225ln

__ 512

__ 522

5ln(524)5ln150 2.ln2

__ 5121ln2

__ 522isundefined.


PC3_SM_Ch02.indd 25 6/23/11 1:45:05 PM

41. 53x5786log53x5log786

3xlog55log786

x5log786_______ 3log5

51.3808;check:53(1.3808)5786

42. 80.2x598.6log80.2x5log98.6

0.2xlog0.85log98.6

x5log98.6________ 0.2log8

511.0391;check:0.80.2(11.0391)598.6

43. 0.80.4x52001log0.80.4x5log2001

0.4xlog0.85log2001

x5 log2001__________ 0.4log0.8

5285.1626;check:0.80.4(285.1626)52001

44. 625x50.007log625x5log0.007

25xlog65log0.007

x5log0.007_________ 25log6

50.5538;check:625(0.5538)50.007

45. 3ex2415510 ex2455__ 3 x245ln5__ 3

x5ln5__ 31454.5108

Check:3eln(5/3)15535__ 315510

46. 42e2x2357235e2x23 nosolution

47. 2e2x15ex2350

ex525______________

252(4)(2)(23)______________________ 4

525___

49___________ 4 5257_______ 4 5

1__ 2or23

1. ex51__ 2 x5ln1__ 2520.6931

2. ex523isnotpossible.

Check:2e2ln(1/2)15eln(1/2)23521__ 4151__ 2235048. 522x232x2250 2x5

3____________

924(5)(22)__________________

2(5)

53___

49_________ 10 51or22__ 5

Alternately,let2x5a ;(5a12)(a21)50a522__ 5ora51 1.2x522__ 5isnotpossible. 2.2x51x50;

Check:5220232022551231225049. a. x M

0 10,000

1 10,700

2 11,449

3 12,250

4 13,108

5 14,026

6 15,007

b.Wheneveryouadd1tox,youmultiplyMby1.07.

c.10,0001.07x527,000 1.07x52.7 xlog1.075log2.7

x5log2.7________

log1.07514.6803yr5176.1640mo;

177mo,or14yr9mo

50. a.Everytimeyouaddoneyear,thepopulationismultipliedby1.0124.

Exponentialfunctionsalwayshavetheaddmultiplyproperty.

b.P(n)5248.71.0124n,withninyearsandtheanswerinmillionsofpeople.

c.248.71.0124n53001.0124n5300______ 248.7

nlog1.01245log3002log248.7

n5log3002log248.7__________________ log1.0124

515.2173yr

515yr79.3419days15yr79days. AroundJune19,2005.Thispredictionisearlierthanthe

actualdateidentifiedbytheU.S.CensusBureau.

Problem Set 2-6Q1. Linear Q2. Exponential

Q3. Inversepower Q4.Quadratic

Q5. Answerswillvary;heightoftides,positiononaFerriswheel,andsoon.

Q6.

t

P

Q7. Parabola Q8.9x2242x149

Q9. 48,96,192 Q10. Exponential

1. a. 14.4____ 3.6557.6____ 14.45

230.4______ 57.65921.6______ 230.454

b. 15a1bln3.655a1bln921.6

45bln921.62bln3.6

5bln921.6_____ 3.6 5bln256b54______

ln256

5 4______ 8ln2

5 1______ 2ln2

50.7213

Substitute0.7213forbintothefirstequation:

15a1 4______ ln256

ln3.6a512ln3.6______ ln256

=0.0760.

Substitutethevaluesforaandbintothegeneralequationy5a1blnx:

y5124ln3.6_______ ln256

1 4______ ln256

lnx50.076010.7213lnx

c.Theequationfitsthedata.

2. a. 10___ 15100____ 105

1000_____ 100510

b. 25a1bln1

55a1bln1000

a52(becauseln150)

Substitute2forainthesecondequation:

5521bln1000b5 3_______ ln1000

5 1_____ ln10

50.4342.

Substitutethevaluesforaandbintothegeneralequationy5a1blnx:

y521 1_____ ln10

lnx5210.4342lnx

c.Theequationfitsthedata.


PC3_SM_Ch02.indd 26 6/23/11 1:45:06 PM

3. a.Theinverseofanexponentialfunctionisalogarithmicfunction.

b.05a1bln100;57305a+bln50;subtractingandsimplifying,

573052bln2b=225730_______ ln2

58,266.6425;

substitutingandsolving,a55730ln100___________ ln2

538,069.2959;y538,069.295928,266.6425lnx

c.y(73.9)538,069.295928,266.6425ln73.952500.30682500yearsold

d.y(20)513,304.647913,300yearsold e.Answerswillvary.

4. a.45a1blog10005a13b;65a1blog1,000,0005a16b;subtracting,253bb52__ 3;45a132__ 35a12a52;m5212__ 3logx

b.m(53109)58.46598.5 m(16031012)511.469411.5

m 3_____ 200050.78400.8 c.212__ 3logx59logx5

3__ 2(922)5

21__ 2

x51021/253.162231010 31.6billiontons

d.False:m(2x)5212__ 3log(2x)521

2__ 3(log21logx)

5212__ 3logx12__ 3log25m(x)12__ 3log2 DoublingtheenergyincreasestheRichtermagnitude

linearlyby2__ 3log250.2006points.Thisisnotsurprising,becauselogarithmicfunctionshavethemultiplyaddproperty.

e.Answerswillvary.

5. a.g(x)56log10x;logbx56log10x

log10x______ log10b

56log10x log10b51__ 6 b510

1/656___

10;

g(x)5log6___

10x5log1.4677x

b.g(x)531lnx50lnx523 x5e2350.0497;h(x)5211lnx50lnx51 x5e15e52.7182Thex-interceptofgise23,andthex-interceptofhise.

6. a.2

b.y52xinvertedisx52y log2x5y,i.e.,y5log2x

orlogx______

log2orlnx____

ln2

c.Thegraphmatchesthedottedfunction.

d.Thisgraphalsomatchesthedottedfunction.

e. Inparametricmode,graphx(t)5f ( t),y(t)5t.

y5f (x)5x329x2123x215:y

x

10

10

5

y5f21(x),givenbyx(t)5t329t2123t215,y(t)5t:

y

x1010

6

7. Domain:x130x23y

x

3

3

3

3

8. Domain:322x0x1.5y

x

3

3

3

3

9. Domain:x20x0y

x

3

3 3

3

10. Domain:x2240(x12)(x22)0x2orx22

y

x

3

3

3

3

11. Domain:3x0x0y

x

3

3

3

3

12. Domain:3x150x25___ 3

10

10

y

x


PC3_SM_Ch02.indd 27 6/23/11 1:45:09 PM

13. a. x y

20.10000 2.8679

0.10000 2.5937

20.01000 2.7319

0.01000 2.7048

20.00100 2.7196

0.00100 2.7169

20.00010 2.7184

0.00010 2.7181

20.00001 2.7182

0.00001 2.7182

b.Thetwopropertiesbalanceout,sothatasxapproaches0,yapproaches2.7182.

c.e52.7182;theyarethesame.


Problem Set 2-7Q1. Addmultiply Q2. Multiplymultiply

Q3. Logarithmic Q4. Multiplyadd

Q5. e Q6. ph5m

Q7. j5log5c Q8. 600deg/s

Q9. y5ax21bx1c,a0 Q10. D

1. a.

5

x

f(x)

5 5

g(x)

y

b.Thegraphsarealmostthesameforlargenegativevaluesofx,butwidelydifferentforlargepositivevaluesofx.

c.Thepointofinflectionisatx50.Thisisfound(onagrapher)astheintersectionofthecurveandtheliney51__ 2c5

1__ 2151__ 2.Thegraphofgisconcaveupforx0

andconcavedownforx0.

d.Asxgrowsverylarge,the1inthedenominatorbecomesinsignificantincomparisontothe2.2x,so

g(x)5 2.2x________

2.2x112.2

x____ 2.2x

51

e.g(x)5 2.2x________

2.2x11 2.2

2x_____ 2.22x

5 1_________ 112.22x

Atableofvaluesshowsthattheexpressionsareequivalent.

2. a. Asxgrowsverylarge,the4inthedenominatorbecomesinsignificantincomparisontoe0.2xso

f (x)5 3e0.2x________

e0.2x143e

0.2x_____ e0.2x

53

b.Pointofinflectionatx56.9314.Thisisfoundgraphicallyastheintersectionofthecurveandtheliney51.5(halfwaybetweentheasymptotesy50andy53).Algebraically,findtheinflectionpointbysolvingf (x)51.5:

f (x)5 3e0.2x________

e0.2x1451.53e0.2x51.5e0.2x16

1.5e0.2x56e0.2x540.2x5ln4x55ln456.9314

c.fisconcaveupforx6.9314andconcavedownforx6.9314.

d.f (x)5 3e0.2x________

e0.2x14e

20.2x______ e20.2x

5 3___________ 114e20.2x

Thegraphscoincide.

3. a. Concaveup

x

y

60 120 180

50

100

150

b.25 1220_________ 11ab20

905 1220__________ 11ab240

212a51220 90190ab24051220a5609 ab240512.555 b24050.0206 b50.0206(1/240) b51.1019

y5 1220____________________ 11(609)(1.1019)2x

c.

60 120 180

400

800

1200y

x

d.y(60)5 1220_____________________ 11(609)(1.1019)260

5435.2804

435students

12105 1220____________________ 11(609)(1.1019)2x

11(609)(1.1019)2x51220_____ 1210

(1.1019)2x5(122/121)21_____________ 609

2xlog(1.1019)5log(122/121)21_____________ 609

x5115.4930min

4. Simulationswillvary.

5. a.Concavedown

50

200

400y

x


PC3_SM_Ch02.indd 28 6/23/11 1:45:11 PM

b.1825 396__________ 11ab210

3945 396__________ 11ab274

1821182ab2105396 3941394ab2745396182ab2105214 394ab27452;

182____ 394b645214____ 2 b

645231.6373

b51.0888;182a1.08882105214a52.7532;

y5 396_________________________ 11(2.7532)(1.0888)2x

c.

50

200

400

y

x

d.y5396____ 2 5198,x5ln2.7532___________ ln1.0888

511.9037

Thepointofinflectionoccursat(11.9037,198).Beforeapproximately12dayspassed,therateofnewinfectionwasincreasing;afterthat,theratewasdecreasing.

e.y5 396__________________________ 11(2.7532)(1.0888)240

5362.7742

After40days,approximately363peoplewereinfected.

f.Answerswillvary.

6. a.f (0)5 1000__________ 11ae2(0)

51000______ 11a5100a59,

sof (x)5 1000_________ 119e2x

Thegraphiscorrect.

b.Thenaturalceilingonthenumberofrabbitsis1000.Ifthepopulationislessthanthis,itwillgrowtowardthislimit.

c.g(0)5 1000__________ 11ae2(0)

51000______ 11a52000a521__ 2,

sog(x)5 1000________ 121__ 2e

2x

Thegraphiscorrect.Thesignofaisnegative,whereasthedefinitionoflogisticfunctionstatesthata0.Sothisisageneralizationofthedefinition.

d. Ifthepopulationisgreaterthanthenumbertheregioncansupport,itwilldecreasetowardthatlimit.

7. a.

5

2

x

f(x)

c 1

c 2

c 3

5

True:cisaverticaldilationfactor.

Takingf (x)5 1_______ 11e2x

astheparentfunction,youget

c_______ 11e2x

5c 1_______ 11e2x5cf (x). b.Changingaseemstotranslatethegraphhorizontally.

5

2

x

f(x)

a 0.2

a 1 a 5

5

Iff (x)5 1_______ 11e2x

,then 1_________ 11ae2x

5 1___________ a1elnae2x

5 1__________ 11elna2x

5 1____________ 11e2(x2lna)

5f(x2lna),

ahorizontaltranslationbylna.Moregenerally,if

f (x)5 c________ 11e2bx

then c_________ 11ae2bx

5 c____________ 11elnae2bx

5 c___________ 11elna2bx

5 c____________ 11e2b x2

lna___ b 5f x2lna____ b ,

ahorizontaltranslationbylna___ b .

c.Horizontaltranslationby3

5

2

x

y

f(x)

g(x)

5

c____________ 11ae2b (x2h)

5f (x2h),ahorizontaltranslationbyh.

d.Youwantanewe

20.4x5aolde20.4(x23)5(1)e20.4x11.25e20.4xe1.2,so

anew5e1.253.3201.Youcanalsofindthisfromthe

resultinpartb:agivesahorizontaltranslationof

lna____ b ,soyouwantlna____ 0.453lna51.2a5e

1.2.

Problem Set 2-8

Review Problems

R0. Journalentrieswillvary.

R1. a.

2 4 6 8 10

10

20

f(x)

x

b. Increasingforx0,decreasingforx0,concaveup

c.Quadraticpowerfunction.Real-worldinterpretationsmayvary.


PC3_SM_Ch02.indd 29 6/23/11 1:45:13 PM

R2. a.y52__ 3x113___ 3.Real-worldinterpretationsmayvary.

b.y

x

y

x

Botharedecreasing.Bothhavethex-axisasanasymptote.Buttheexponentialfunctioncrossesthey-axis,whereastheinversefunctionhasnoy-intercept(andhasthey-axisasanasymptote).

c.They-interceptisnonzero.

y568__ 3x22___ 3 56 3

___ 9__ 643

__ 8__ 3x5(3.1201)(1.3867)x

Real-worldinterpretationsmayvary.

d.y521.2x219x12;Thecoefficientofx2isnegative,whichindicatesthegraphisconcavedown.Real-worldinterpretationsmayvary.

e.Vertex(5,3);y-intercept:y52(0)25

213553

R3. a.Addmultiplyexponential;f (x)5483___

0.5x

b.Multiplymultiplypower(inversevariation);g(x)572x21

c.Addaddlinear;h (x)52x118 d.Constant-second-differencesquadratic;

q(x)5x2213x154

e. i. f (12)52131__ 3

ii. f (12)5160

iii. f (12)5180

f.f (x1c)5531.3x1c5531.3x1.3c51.3cf (x)R4. a.Anexponent

b.p5log10z

c. 101.4771530

d.Answerswillvary.Sampleanswers:

i. log(10010)5log100053log1001log10521153

ii. log10,000_______ 1,0005log1051

log10,0002log1,000542351

iii. log1035log100053 3log1053153

e.60

R5. a. cp5m

b. log73051.7478

c.63

d. log(x11)1log(x22)51 log[(x11)(x22)]511015(x11)(x22)105x22x2205x22x212(x13)(x24)501.x523 or 2. x54Check:1.log(22)1log(25)isundefined.2.log51log25log1051 x54

e. 32x2157x(2x21)log35xlog7 2x21_______ x 5

log7_____

log3221__ x5

log7_____

log3

22log7_____ log3

51__ x

x5 1________ 22

log7____ log354.3714

Check:32(4.3714)2154946.712974.371454946.7129

R6. a. f1(x)andf2(x)arereflectionsofeachotheracrosstheliney5x.

x

y

f1

5

5(x)

f2(x)

b.f (x)55e20.4x55e20.4x550.6703x

g(x)54.37.4x7.45ebb5ln7.452.0014g(x)54.3e2.0014x

c.Multiplyaddpropertyy5213log2

x____ 100

d.

25 50 75 100

20

40

60

x

y

e.y5213log21____ 100

586.3701ftdeep(byextrapolation).

R7. a.

x

y

f(x)

g(x)

5 10 15 205

5

10

15

b.Whenxisalargenegativenumber,thedenominatoroff (x)isessentiallyequalto10,soforlargenegativex,

f (x)5102x_______

2x110102

x_______ 10 52x5g(x).

Butforlargepositivex,the10inthedenominatoroff (x)isnegligiblecomparedwiththe2x;so

f (x)5102x_______

2x110102

x_______ 2x 510.

c.f (x)5102x_______

2x1102

2x____ 22x

5 10____________ 111022x

d.g(x)52x5e(ln2)x


PC3_SM_Ch02.indd 30 6/23/11 1:45:16 PM

e.Thesizeofthepopulationwouldbelimitedbythecapacityoftheisland.

755 460_________ 11ab26

3555 460__________ 11ab224

75175ab265460 3551355ab2245460

75ab265385 355ab2245105

75ab26__________

355ab2245385____ 105

b18511___ 371___ 15517.3555

b5(17.3555)1/1851.1718

a5 385______________ 75(1.1718)26

513.2906

f (x)5 460_________________________ 11(13.2906)(1.1718)2x

f (12)5 460__________________________ 11(13.2906)(1.1718)212

5154.2335

f (18)5 460__________________________ 11(13.2906)(1.1718)218

5260.5072

20 40

200

400

x

y

4375 460_________________________ 11(13.2906)(1.1718)2x

1.17182x5460___ 43721__________ 13.290650.0039

2xln1.17185ln0.0039

x52ln0.0039_____________ ln1.1718

534.8878months

Concept Problems

C1. a.

4

10

7

1

5

1

3

1

11

111

3

1

5

1

7

x

y

4

b.Vertexat(2,25)

x

7

1

5

1

3

1

11

111

31

5

1

7

y

c.y

x

0.30.9

1.5

2.1

0.30.9

1.5

2.1

2.7

3.3

C2. a.f (9)520.7119;g(60)5324

b.Thegraphslooklinear.

x1

2

5

20

50

200

500

10

100

1000

0 5 10 15

h(x)


PC3_SM_Ch02.indd 31 6/23/11 1:45:18 PM

x1

1 10 100

10

2

5

20

50

200

500

100

1000p(x)

c.f (x)510000.65xlogf (x)5log(10000.65x)logf (x)5log10001log0.65xlogf (x)5log10001xlog0.65y-interceptislog1000;slopeislog0.65.Thegraphislinear.

d.g(x)50.09x2logg(x)5log(0.09x2)logg(x)5log0.091logx2logg(x)5log0.0912logxy-interceptislog0.09;slopeis2

C3. a. i.4005300C11000____________ C11 C56

y5180011000e0.7x________________

61e0.7x

ii.13005300C11000____________ C11 C520.3

y529011000e0.7x_______________

20.31e0.7x

iii.2995300C11000____________ C11 C52701

y52210,30011,000e0.7x

_____________________ 27011e0.7x

b.

5

400

1000

1300

x

y

c 6

c 0.3

c 701

Answerswillvary,seepartd.

c.Thegraphsfollowthedirectionofthelinesegments.

x

y

d. If400treesareplanted,thepopulationincreasesatfirstandthenlevelsoffat1000.If1300(toomany)treesareplanted,thepopulationdecreasestoleveloffat1000.If299(toofew)treesareplanted,thepopulationdwindlesuntilalltreesaredead.

e.

x

y

f.Youcandrawthegraphfollowingthedirectionofthelinesegmentstogetanideaofwhathappensatdifferentinitialconditions.


PC3_SM_Ch02.indd 32 6/23/11 1:45:19 PM

Chapter Test

T1. a. y5ax1b

b. y5ax21bx1c,a0

c. y5axb,a0

d. y5aebxory5abx,a,b0,b0andb1inthecaseofy5abx

e. y5a1blogcx,b0andc0,c1

f. y5 c_________ 11ae2bx

ory5 c_________ 11ab2x

,a,b,c0,b0and

b1inthecaseofy5 c_________ 11ab2x

T2. a. Logarithmic

b. Exponential

c. Logistic

d. Quadratic

e. Power

f. Linear

T3. a. Addadd

b. Constant-second-differences

c. Multiplymultiply

d. Addmultiply

e. Multiplyadd

T4. ac5b

T5. log5x5xlog5 T6. 8

T7. 45

T8. 4x232x2450(2x)223(2x)2450Let2x5a.a223a2450(a24)(a11)50a54or21,so2x54or2x521x52ornosolutionCheck:422322245162122450

T9. log2(x24)2log2(x13)58

log2(x24)_______ (x13)

58x24_____ x135285256

x245256x17682255x5772x523.0274,whichcannotbe,sotherearenosolutions.

T10. f (10)_____

f(5)5600____ 75585

4800_____ 6005f (20)_____

f (10)

T11. f (x)5axb;a5b575anda10b5600 10___ 5

b5600____ 7558b53;a1035600a50.6;

f (x)50.6x3

T12. 0.6(15)352025,0.6(20)354800;thefunctioniscorrect.

T13. f (100)5600,000lb5300tons

T14. f (x)50.6x353000x355000x510 3__

5517.0997ft

T15.

10 20 30

50

100

x

g (x)

Graphwillbeconcaveup.Thefunctionappearstostartatapositivenumber,decreaserapidly,thenleveloffasxgrowslarge.Alinearfunctioncannotwork,becausethegraphappearstobeconcave.Also,aninversevariationpowerfunctioncannotwork,becauseitappearsthatthegraphwillintersecttheverticalaxis.

T16. 94.85ab3,40.85ab11

40.8____ 94.85ab11____ ab3

b850.4303b50.8999

a5 94.8_________ 0.89993

5130.0510

f (x)5(130.0510)(0.8999)x;f (5)5(130.0510)(0.8999)5576.7840Ff (7)5(130.0510)(0.8999)7562.1919Ff (9)5(130.0510)(0.8999)9550.3729F

T17. f (0)5(130.0510)(0.8999)05130.0510Faboveroomtemperature.

T18. f (30)5(130.0510)(0.8999)3055.5088Faboveroomtemperature.

T19. y5713xlogy5log(713x)5log71log13x

5log71(log13)xT20. Thegraphwillbeconcavedown.Aquadraticfunctionmight

fitthedata.

1 2 3 4 5 6 71

100

200

t

h

T21. Thefirstdifferencesare216216655023422165182202234521417422205246

Theseconddifferencesare18250523221421852322462(214)5232

T22. 166=4a12b1c216=9a13b1c234=16a14b1c

abc

5

491623

411

1

21

166216234

5

216130230

h(t)5216t21130t230 h(5)5216(52)1130(5)2305220,whichagrees. h(6)5216(62)1130(6)2305174,whichagrees.


PC3_SM_Ch02.indd 33 6/23/11 1:45:20 PM

T23. f (18)55.5,f (54)56.2 4.85a1bln6 4.15a1bln2 Bysubtraction, 0.75(ln62ln2)b51.0986b b50.6371,a54.12(0.6371)(ln2)53.6583 y53.658310.6371lnx

T24. 3635ab2,8305ab11

830____ 3635ab

11____ ab2

2.28655b9b51.0962

a5 363_________ 1.09622

5302.0582

f (x)5(302.0582)(1.0962)x;f (5)5478.2229;f (7)5574.7067

T25. g(2)5362.0488;g(5)5484.0232;g(7)5583.2807g(11)5829.2796

T26.

50

2000

y

x

f

g

T27. Thelogisticfunctionismorereasonablebecausethetowncanholdonlyalimitednumberofpeople.



PC3_SM_Ch02.indd 34 6/23/11 1:45:21 PM

Problem Set 3-1 1. Yes,y52.1x1 3.4.

2.

5 10

10

20

y

x

Theregressionlinesmatch.

3. y(14)52.1(14)13.4532.8; 33sit-ups. Explanationsmayvary.Fourteendaysisanextrapolation

fromthegivendata,andextrapolatingfrequentlygivesincorrectpredictions.

4. y 5 2.1x 1 3.4 y 2 y (y 2 y)

7.6 0.4 0.16

11.8 21.8 3.24

16.0 3.0 9.00

20.2 22.2 4.84

24.4 0.6 0.36

5. SSres517.60

6. y 5 2.1x 1 3.5 y 2 y (y 2 y) 2

7.7 0.3 0.09

11.9 21.9 3.61

16.1 2.9 8.41

20.3 22.3 5.29

24.5 0.5 0.25

SSres517.65

y 5 2.1x 1 3.4 y 2 y (y 2 y) 2

7.8 0.2 0.04

12.2 22.2 4.84

16.6 2.4 5.76

21.0 23 9.00

25.4 20.4 0.16

SSres519.80

Problem Set 3-2Q1. 12 Q2.23

Q3. 30 Q4. Itequals1or21.

Q5. Power Q6. Exponential

Q7. 35 Q8.56

Q9. B Q10. m2x212bmx1b2

1. a. Agraphingcalculatorgivesy5 1.4x13.8,withr5 0.9842.

b.

10 20 30

20

40

y

x

Thelinefitsthedatawell.

c.n510,x5185,__

x518.5,y5297,__

y5 29.7y(

__ x)5y(18.5)51.4(18.5)13.8529.75

__ y

d.

SSdev51502.10,SSres546.80,

r25SSdev2 SSres____________ SSdev

5 0.9688 Takethepositivesquare

rootbecausetheregressionlinehaspositiveslope.

r5 _______

0.968850.9842,whichagreeswithparta.

e.

10 20 30

20

40

y 1.4x 3.8

y 1.5x 1.95

y

x

Itishardtotellwhichlinefitsbetter.

y 5 1.5x 1 1.95 y 2 y (y 2 y) 2

9.45 1.55 2.4025

13.95 2.05 4.2025

18.45 0.55 0.3025

22.95 4.05 16.4025

27.45 22.45 6.0025

31.95 22.95 8.7025

36.45 23.45 11.9025

40.95 1.05 1.1025

45.45 21.45 2.1025

49.95 1.05 1.1025

SSres5 54.2250,whichislargerthanSSresfortheregressionline.

y 2 __

y (y 2 __

y)2 y 2 y (y 2 y) 2

218.7 349.69 0.2 0.04

213.7 187.69 1.0 1.00 210.7 114.49 20.2 0.04 22.7 7.29 3.6 12.96 24.7 22.09 22.6 6.76 20.7 0.49 22.8 7.84 3.3 10.89 23.0 9.00

12.3 151.29 1.8 3.24

14.3 204.49 20.4 0.16

21.3 453.69 2.4 5.76

1502.10 46.80


Chapter 3 Fitting Functions to Data

PC3_SM_Ch03.indd 35 6/23/11 1:59:33 PM

2. a. y567.6358x1 26,139.5007,r50.9595

10 20 30

100

200

y (1000 dollars)

x (100 ft2)

Thelinefitswellbecausethepointsclusternearit.

b. y(5,000)5364,318.6490$364,000

1,000,0002 26,139.5007 _________________________

67.6358 514,398.588714,400ft2

Outside:extrapolation;inside:interpolation

c.Alotcostsabout$26,140.Ahousecostsabout$67.64perft2.

d. __

x52470ft2,__

y5$193,200 y(

__ x)5y(2470)

567.6358(2470)126,139.5007 5 $193,2005__

y

10 20 30

100

200

y (1000 dollars)

x (100 ft2)

y $193,200

x 2470 ft2

e. y 2 __

y (y 2 __

y)2

238,200 1,459,240,000

225,200 635,040,000

23,200 10,240,000

24,200 17,640,000

13,800 190,440,000

1,800 3,240,000

5,800 33,640,000

5,800 33,640,000

16,800 282,240,000

26,800 718,240,000

SSdev53,383,600,000

y 2 y (y 2 y)2

352.4229 124,201.9057

2174.7430 305,535.1247 1,534.5080 2,354,715.0363

26,229.0748 38,801,373.983611,770.9251 138,554,677.9482

26,992.6578 48,897,263.892422,992.6578 8,956,001.043622,992.6578 8,956,001.0436 1,243.7591 1,546,936.8920

4,480.1762 20,071,978.8856

SSres5268,293,685.7563

r 2 5 3,383,600,0002268,293,685.7563__________________________________ 3,383,600,000

5 0.9207

r5 0.9595,whichagreeswithparta.

f.Twohousescanhavethesamesquarefootage.Twohousescanhavethesameprice.Therelationshipbetweensquarefootageandpriceisonlystatistical,notenforced,andthepriceisinfluencedbyfactorsotherthansquarefootage.

3. a. y520.05x117,r251,r521,whichmeansaperfectfit.risnegativebecausetheremaininggasdecreasesasthedistancedrivenincreases.

b. __

y515.18gal

y 2 __

y (y 2 __

y)2 y 2 y (y 2 y)2

1.52 2.3104 0 0

0.72 0.5184 0 0

20.38 0.1444 0 0

20.68 0.4624 0 0

21.18 1.3924 0 0

SSdev54.828,SSres5 0

r254.82820__________ 4.828 51,r521,whichagreeswithparta.

c.

20 40 60

5

10

15

y

x

Datapointsareallontheline.

d.Atx50mi,thetankholdsy(0)517gal;thecargets20mi/gal.

e. y(340)520.0534011750gal;notveryconfident,becausedrivingconditionscouldchange.

4. a. y(40)559.0(4.0)13555591.Becausethescoresaresoscattered,thismaynotbethemostreliableprediction.

b. r1 ____

0.1450.3741Positive,becausetheslope(59.0)ispositive.

5. a.y

xr 0.95


PC3_SM_Ch03.indd 36 6/23/11 1:59:35 PM

b.y

xr 0.8

c.y

x

r 0.7

d.y

xr 0

6. Answerswillvary.

Problem Set 3-3Q1. Exponential Q2. Inversepower

Q3. Cubic Q4. Logistic

Q5. Periodicmeansrepeatingoveradefinedinterval,orperiod.

Q6. Answerswillvary.

Advertising ($)

Sales (units)

Q7. y5x2 Q8.81x2272x116

Q9. 12,6,3 Q10. Power

1. a.Bothapowerfunctionandanexponentialfunctionhavetheproperrightendpointbehavior:increasingtoinfinity.Onlyanexponentialfunctionhasthecorrectleftendpointbehavior:beingnonzero.

b. y5346.92911.4972x,withr50.9818

5

y

x

10,000

c. y(0)5346.9291347bacteriay(24)55,584,729.3315 5.6millionbacteria

d. y5abx5 100,000

loga1xlogb5log100,00055

x552loga_________ logb

552log346.9291

__________________ log1.4972

5 14.033114.0h

Check:y(14.0331)5100,000bacteria.

2. a.Yes,byusingmarginswhosewidthtotaledthepagewidth.Inthatcase,theparagraphwouldbeinfinitelylong,matchingadecreasingpowerfunction,ratherthanfinitelylongasadecreasingexponentialfunctionwouldsuggest.

Power:y537.2746x21.0697,r520.9992

Exponential:y5 41.17760.7025x,r520.9674

b.

4 8

30

y

x

Power

Exponential

Notonlydoestheexponentialfunction(dotted)havethewrongleftendpointbehavior,butitalsomissesmoreofthedatapoints.


PC3_SM_Ch03.indd 37 6/23/11 1:59:37 PM

c.A5xy__ 7

x y A

6.5 5 4.64285.5 6 4.71425 7 5

3.75 9 4.82143 11 4.71421.5 24 5.14281 38 5.4285

y520.1090x15.3325, r520.7883

y

x

5

5

3. a.

2000

10

20

y

x

Concavedown.Thegraphdecreasesmoresteeply

(presumablyto2)towardx50,hasapositivex-intercept,andincreaseslesssteeplytotheright.

b. y5 2138.1230119.9956lnx;r50.9999999799,whichisnearly1.

c.

2000

10

20

y

x

d. y(2500)518.323618.32yr

1386121.97_____________ 2 517.915yr

e. y(5000)532.1835 32.18yr Extrapolation,because50003000. Extrapolationisprobablysafeinthiscase,becausethe

bankprobablyusesasimpleformulatocalculateinterest.Thiswouldmaketheregressionequationapplyforallvaluesofx.

4. a.0yearsPower:y50.0005521x1.4980,r50.999998Exponential:y51.07631.0013x,r50.8951

b.

5000

100

200

y

x

Power

Exponential

Thepowerfunctionfitsverywell.

c.

200

100

200

y

x

(period)

(mass)

Thescatterplothasverylittleshape.Noneoftheregressiontypesavailableonagraphingcalculatorgivesagraphwhoseshapematchesthedata.

d. y(430)54.86484.86yr e.Answerswillvary.Keplersthirdlawstatesthattheperiod

ofaplanetsorbitisproportionaltothe3__ 2powerofitsdistancefromtheSun,andtheregressionequation(witha1.4980power)agreeswiththatmodelveryclosely.

5. a.Growthisbasicallyexponential,butphysicallimitseventuallymakethepopulationleveloff.Alogisticfunctionfitsdatathathaveasymptotesatbothendpointsbutareexponentialinthemiddle.

y5 327.5140______________________ 1110.0703e20.4029x

5 10

100

200

300y

x

b. y(20)5326.4745 326roadrunnersy 327.5140 328roadrunnersasx Theinflectionpointappearstobeatx5.5yr.


PC3_SM_Ch03.indd 38 6/23/11 1:59:40 PM

c. __

y5158.5roadrunners

y 2 __

y (y 2 __

y)2 y2y (y2y)2

2128.5 16,512.25 0.4153 0.17252114.5 13,110.25 1.6330 2.66692100.5 10,100.25 21.5676 2.4574277.5 6,006.25 20.7496 0.5619248.5 2,352.25 1.1633 1.3533220.5 420.25 21.7936 3.217116.5 272.25 2.3942 5.7325

44.5 1,980.25 21.7203 2.9596

75.5 5,700.25 0.2077 0.0431

101.5 10,302.25 1.6919 2.8628

117.5 13,806.25 21.7752 3.1515

134.5 18,090.25 0.4914 0.2414

SSdev598,653.00 SSres525.4205

r25 SSdev2SSres____________ SSdev

5

98,653.00225.4205_____________________ 98,653.00 5 0.9997,

whichisverycloseto1.

6. a. Increasedcompetitionforresources(food,space,etc.)limitsthepossibilityofpopulationgrowthbecausethenumberofdeathsincreasesfasterthanthenumberofbirths.

b. y 520.0012x210.3890x12.9313,R250.8536

200

20

y

x

c. y(400)5242.6474243roadrunners/yr.Thepopulationishigherthancanbesupportedandwouldbeexpectedtofallbyabout43roadrunnersoverthenextyear,becauseofdeathsoutnumberingbirths.

d. __

y523.___

90roadrunners/yr

y 2 __

y (y 2 __

y)2 y2y (y2y)2

29.9090 98.1900 0.5300 0.2809

29.9090 98.1900 23.6134 13.0571

20.9090 0.8264 1.7359 3.0134

5.0909 25.9173 2.8085 7.8881

4.0909 16.7355 22.5080 6.2901

13.0909 171.3719 4.3310 18.7579

4.0909 16.7355 24.5004 20.2542

7.0909 50.2809 0.9154 0.8381

2.0909 4.3719 0.8900 0.7922

27.9090 62.5537 23.0742 9.4508

26.9090 47.7355 2.4850 6.1753

SSdev5592.9090 SSres586.7985

R25 SSdev2SSres____________ SSdev

5 592.9090286.7985_____________________

592.9090 5 0.8536,

asinpartb.

Problem Set 34 Q1.

Q2.

Q3.

Q4.

Q5.

Q6. 57535

Q7.184356

Q8. 72549

Q9. Addmultiply

Q10. Multiplyadd


PC3_SM_Ch03.indd 39 6/23/11 1:59:42 PM

1. a. x y

1 3

3 12

5 48

7 192

9 768

b.,c.

x

y

100

50

20

5

2

1000

500

200

10

51

010 15

2. a. x y

2 288

4 103.68

6 37.3248

8 13

paul a foerster precalculus with trigonometry concepts and applications

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