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paul a foerster precalculus with trigonometry concepts and applications textbook for
Citation preview
Third Edition
P A U L A . F O E R S T E R
SOLUTIONS MANUAL
PC3_SM_FM.indd 1 6/23/11 3:54:48 PM
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
PC3_SM_FM.indd 2 6/23/11 3:54:48 PM
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
Precalculus with Trigonometry: Solutions Manual Problem Set 1-2 3 2012 Key Curriculum Press
PC3_SM_Ch01.indd 3 6/23/11 1:40:24 PM
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)
4 Problem Set 1-3 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
PC3_SM_Ch01.indd 4 6/23/11 1:40:28 PM
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
Precalculus with Trigonometry: Solutions Manual Problem Set 1-3 5 2012 Key Curriculum Press
PC3_SM_Ch01.indd 5 6/23/11 1:40:31 PM
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
6 Problem Set 1-4 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
PC3_SM_Ch01.indd 6 6/23/11 1:40:33 PM
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 1-4 7 2012 Key Curriculum Press
PC3_SM_Ch01.indd 7 6/23/11 1:40:37 PM
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 .
8 Problem Set 1-4 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
PC3_SM_Ch01.indd 8 6/23/11 1:40:39 PM
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
Precalculus with Trigonometry: Solutions Manual Problem Set 1-5 9 2012 Key Curriculum Press
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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.
10 Problem Set 1-5 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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10. a.
5
10
y
x5 5
b. Notafunction
c. Graphergraphagreeswithgraphonpaper.
11. a.
5
y
x
10
10 5
5
5
b. Notafunction
c. Graphergraphagreeswithgraphonpaper.
12. a.
5
y10
5 510x
5
b. Function
c. Graphergraphagreeswithgraphonpaper.
13. a.
5
y
x
10
10
5
5
b. Function
c. Graphergraphagreeswithgraphonpaper.
14. a.
5
10y
x105
5
b. Notafunction
c. Graphergraphagreeswithgraphonpaper.
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 1-5 11 2012 Key Curriculum Press
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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
12 Problem Set 1-5 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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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.
Precalculus with Trigonometry: Solutions Manual Problem Set 1-5 13 2012 Key Curriculum Press
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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
14 Problem Set 1-6 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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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
Precalculus with Trigonometry: Solutions Manual Problem Set 1-6 15 2012 Key Curriculum Press
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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
14. Answerswillvary.
16 Problem Set 1-6 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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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
Precalculus with Trigonometry: Solutions Manual Problem Set 1-8 17 2012 Key Curriculum Press
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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.
18 Problem Set 1-8 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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Precalculus with Trigonometry: Solutions Manual Problem Set 1-8 19 2012 Key Curriculum Press
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
20 Problem Set 1-8 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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.)
T29. Answerswillvary.
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 2-2 21 2012 Key Curriculum Press
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
22 Problem Set 2-3 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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
6. Addaddproperty:linear
7. Multiplymultiplyproperty:power,inversevariation
8. Addaddproperty:linear
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 2-3 23 2012 Key Curriculum Press
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.
24 Problem Set 2-4 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 2-5 25 2012 Key Curriculum Press
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.
26 Problem Set 2-6 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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
Precalculus with Trigonometry: Solutions Manual Problem Set 2-6 27 2012 Key Curriculum Press
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.
14. Answerswillvary.
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
28 Problem Set 2-7 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 2-8 29 2012 Key Curriculum Press
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
30 Problem Set 2-8 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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)
Precalculus with Trigonometry: Solutions Manual Problem Set 2-8 31 2012 Key Curriculum Press
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.
32 Problem Set 2-8 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 2-8 33 2012 Key Curriculum Press
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.
T28. Answerswillvary.
34 Problem Set 2-8 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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
Precalculus with Trigonometry: Solutions Manual Problem Set 3-2 35 2012 Key Curriculum Press
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 $193,200
x 2470 ft2
e. y 2 __
y (y 2 __
y)2
238,200 1,459,240,000
225,200 635,040,000
23,200 10,240,000
24,200 17,640,000
13,800 190,440,000
1,800 3,240,000
5,800 33,640,000
5,800 33,640,000
16,800 282,240,000
26,800 718,240,000
SSdev53,383,600,000
y 2 y (y 2 y)2
352.4229 124,201.9057
2174.7430 305,535.1247 1,534.5080 2,354,715.0363
26,229.0748 38,801,373.983611,770.9251 138,554,677.9482
26,992.6578 48,897,263.892422,992.6578 8,956,001.043622,992.6578 8,956,001.0436 1,243.7591 1,546,936.8920
4,480.1762 20,071,978.8856
SSres5268,293,685.7563
r 2 5 3,383,600,0002268,293,685.7563__________________________________ 3,383,600,000
5 0.9207
r5 0.9595,whichagreeswithparta.
f.Twohousescanhavethesamesquarefootage.Twohousescanhavethesameprice.Therelationshipbetweensquarefootageandpriceisonlystatistical,notenforced,andthepriceisinfluencedbyfactorsotherthansquarefootage.
3. a. y520.05x117,r251,r521,whichmeansaperfectfit.risnegativebecausetheremaininggasdecreasesasthedistancedrivenincreases.
b. __
y515.18gal
y 2 __
y (y 2 __
y)2 y 2 y (y 2 y)2
1.52 2.3104 0 0
0.72 0.5184 0 0
20.38 0.1444 0 0
20.68 0.4624 0 0
21.18 1.3924 0 0
SSdev54.828,SSres5 0
r254.82820__________ 4.828 51,r521,whichagreeswithparta.
c.
20 40 60
5
10
15
y
x
Datapointsareallontheline.
d.Atx50mi,thetankholdsy(0)517gal;thecargets20mi/gal.
e. y(340)520.0534011750gal;notveryconfident,becausedrivingconditionscouldchange.
4. a. y(40)559.0(4.0)13555591.Becausethescoresaresoscattered,thismaynotbethemostreliableprediction.
b. r1 ____
0.1450.3741Positive,becausetheslope(59.0)ispositive.
5. a.y
xr 0.95
36 Problem Set 3-2 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
PC3_SM_Ch03.indd 36 6/23/11 1:59:35 PM
b.y
xr 0.8
c.y
x
r 0.7
d.y
xr 0
6. Answerswillvary.
Problem Set 3-3Q1. Exponential Q2. Inversepower
Q3. Cubic Q4. Logistic
Q5. Periodicmeansrepeatingoveradefinedinterval,orperiod.
Q6. Answerswillvary.
Advertising ($)
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.
Precalculus with Trigonometry: Solutions Manual Problem Set 3-3 37 2012 Key Curriculum Press
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.
38 Problem Set 3-3 Precalculus with Trigonometry: Solutions Manual 2012 Key Curriculum Press
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
Precalculus with Trigonometry: Solutions Manual Problem Set 3-4 39 2012 Key Curriculum Press
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