Calculus solution manual 9th 9th edition adams

FOREWORD

Thesesolutionsareprovidedforthebeneﬁtofinstructorsusingthetextbooks: Calculus:ACompleteCourse(9thEdition), Single-VariableCalculus(9thEdition), and CalculusofSeveralVariables(9thEdition)

byR.A.AdamsandChrisEssex,publishedbyPearsonCanada.Forthemostpart,thesolutionsaredetailed,especiallyinexercisesoncorematerialandtechniques.Occasionallysome detailsareomitted—forexample,inexercisesonapplicationsofintegration,theevaluationof theintegralsencounteredisnotalwaysgivenwiththesamedegreeofdetailastheevaluationof integralsfoundinthoseexercisesdealingspeciﬁcallywithtechniquesofintegration.

Instructorsmaywishtomakethesesolutionsavailabletotheirstudents.However,students shouldusesuchsolutionswithcaution.Itisalwaysmorebeneﬁcialforthemtoattemptexercisesandproblemsontheirown,beforetheylookatsolutionsdonebyothers.Iftheyexamine solutionsas“studymaterial”priortoattemptingtheexercises,theycanlosemuchofthebeneﬁtthatfollowsfromdiligentattemptstodeveloptheirownanalyticalpowers.Whentheyhave triedunsuccessfullytosolveaproblem,thenlookingatasolutioncangivethema“hint”for asecondattempt.Separate StudentSolutionsManuals forthebooksareavailableforstudents. Theycontainthesolutionstotheeven-numberedexercisesonly. November,2016.

R.A.Adams adms@math.ubc.ca

ChrisEssex essex@uwo.ca

SolutionsforChapterP 1

SolutionsforChapter1

SolutionsforChapter14 538

SolutionsforChapter15 579

SolutionsforChapter16 610

SolutionsforChapter17 637

SolutionsforChapter18 644

SolutionsforChapter18-cosv9 671

SolutionsforAppendices 683

NOTE:“SolutionsforChapter18-cosv9”isonlyneededbyusersof CalculusofSeveralVariables (9thEdition),whichincludesextramaterialinSections18.2and18.5thatisfoundin Calculus:aCompleteCourse andin Single-VariableCalculus inSections7.9and3.7respectively. SolutionsforChapter18-cosv9containsonlythesolutionsforthetwoSections18.2and18.9in theSeveralVariablesbook.AllotherSectionsarein“SolutionsforChapter18.”

ItshouldalsobenotedthatsomeofthematerialinChapter18isbeyondthescopeofmost studentsinsingle-variablecalculuscoursesasitrequirestheuseofmultivariablefunctionsand partialderivatives.

CHAPTERP.PRELIMINARIES

SectionP.1RealNumbersandtheRealLine (page10)

1. 2 9 D 0:22222222 D 0:2

2. 1 11 D 0:09090909 D 0:09

3. If x D 0:121212 ,then 100x D 12:121212 D 12 C x Thus 99x D 12 and x D 12=99 D 4=33.

4. If x D 3:277777 ,then 10x 32 D 0:77777 and 100x 320 D 7 C .10x 32/,or 90x D 295.Thus x D 295=90 D 59=18

5. 1=7 D 0:142857142857 D 0:142857

2=7 D 0:285714285714 D 0:285714

3=7 D 0:428571428571 D 0:428571

4=7 D 0:571428571428 D 0:571428 note thesamecyclicorderoftherepeatingdigits

5=7 D 0:714285714285 D 0:714285

6=7 D 0:857142857142 D 0:857142

6. Two differentdecimalexpansionscanrepresentthesame number.Forinstance,both 0:999999 D 0:9 and 1:000000 D 1:0 representthenumber1.

7. x 0 and x 5 deﬁnetheinterval Œ0;5

8. x<2 and x 3 deﬁnetheinterval Œ 3;2/.

9. x> 5 or x< 6 deﬁnestheunion . 1; 6/ [ . 5; 1/

10. x 1 deﬁnestheinterval . 1; 1

11. x> 2 deﬁnestheinterval . 2; 1/.

12. x<4 or x 2 deﬁnestheinterval . 1; 1/,thatis,the wholerealline.

13. If 2x>4,then x< 2.Solution: . 1; 2/

14. If 3x C 5 8,then 3x 8 5 3 and x 1.Solution: . 1;1

15. If 5x 3 7 3x,then 8x 10 and x 5=4.Solution: . 1;5=4

16. If 6 x 4 3x 4 2 ,then 6 x 6x 8.Thus 14 7x and x 2.Solution: . 1;2

17. If 3.2 x/<2.3 C x/,then 0<5x and x>0.Solution: .0; 1/

18. If x2 <9,then jxj <3 and 3<x<3.Solution: . 3;3/

19. Given: 1=.2 x/<3.

CASEI.If x<2,then 1<3.2 x/ D 6 3x,so 3x<5 and x<5=3.Thiscasehassolutions x<5=3

CASEII.If x>2,then 1>3.2 x/ D 6 3x,so 3x>5 and x>5=3.Thiscasehassolutions x>2

Solution: . 1;5=3/ [ .2; 1/

20. Given: .x C 1/=x 2

CASEI.If x>0,then x C 1 2x,so x 1

CASEII.If x<0,then x C 1 2x,so x 1.(not possible)

Solution: .0;1

21. Given: x2 2x 0.Then x.x 2/ 0.Thisisonly possibleif x 0 and x 2.Solution: Œ0;2

22. Given 6x2 5x 1,then .2x 1/.3x 1/ 0,soeither x 1=2 and x 1=3,or x 1=3 and x 1=2.Thelatter combinationisnotpossible.Thesolutionsetis Œ1=3;1=2

23. Given x3 >4x,wehave x.x2 4/>0.Thisispossible if x<0 and x2 <4,orif x>0 and x2 >4.The possibilitiesare,therefore, 2<x<0 or 2<x< 1

Solution: . 2;0/ [ .2; 1/

24. Given x2 x 2,then x2 x 2 0 so .x 2/.x C1/ 0. Thisispossibleif x 2 and x 1 orif x 2 and x 1.Thelattersituationisnotpossible.Thesolution setis Œ 1;2.

25. Given: x 2 1 C 4 x

CASE I.If x>0,then x2 2x C 8,sothat x2 2x 8 0,or .x 4/.x C 2/ 0.Thisispossiblefor x>0 onlyif x 4

CASEII.If x<0,thenwemusthave .x 4/.x C 2/ 0, whichispossiblefor x<0 onlyif x 2

Solution: Œ 2;0/ [ Œ4; 1/

26. Given: 3 x 1 < 2 x C 1 . CASE I.If x>1 then .x 1/.x C 1/>0,sothat 3.x C 1/<2.x 1/.Thus x< 5.Therearenosolutions inthiscase.

CASEII.If 1<x<1,then .x 1/.x C 1/<0,so 3.x C 1/>2.x 1/.Thus x> 5.Inthiscaseall numbersin . 1;1/ aresolutions.

CASEIII.If x< 1,then .x 1/.x C 1/>0,sothat 3.x C 1/<2.x 1/.Thus x< 5.Allnumbers x< 5 aresolutions.

Solutions: . 1; 5/ [ . 1;1/.

27. If jxjD 3 then x D˙3

28. If jx 3jD 7,then x 3 D˙7,so x D 4 or x D 10

29. If j2t C 5jD 4,then 2t C 5 D˙4,so t D 9=2 or t D 1=2

30. Ifj1 t jD 1,then 1 t D˙1,so t D 0 or t D 2

31. If j8 3sjD 9,then 8 3s D˙9,so 3s D 1 or 17,and s D 1=3 or s D 17=3

32. If ˇ ˇ ˇ s 2 1ˇ ˇ ˇ D 1,then s 2 1 D˙1,so s D 0 or s D 4

33. If jxj <2,then x isin . 2;2/

34. If jxj 2,then x isin Œ 2;2.

35. If js 1j 2,then 1 2 s 1 C 2,so s isin Œ 1;3

36. If jt C 2j <1,then 2 1<t< 2 C 1,so t isin . 3; 1/.

37. If j3x 7j <2,then 7 2<3x<7C2,so x isin .5=3;3/.

38. If j2x C 5j <1,then 5 1<2x< 5 C 1,so x isin . 3; 2/.

39. If ˇ ˇ ˇ x 2 1ˇ ˇ ˇ 1,then 1 1 x 2 1 C 1,so x is in Œ0;4.

40. If ˇ ˇ ˇ 2 x 2 ˇ ˇ ˇ < 1 2 ,then x=2 liesbetween 2 .1=2/ and 2 C .1=2/.Thus x isin .3;5/

41. Theinequality jx C 1j > jx 3j saysthatthedistance from x to 1 isgreaterthanthedistancefrom x to3,so x mustbetotherightofthepointhalf-waybetween 1 and3.Thus x>1.

42. jx 3j <2jxj, x2 6x C 9 D .x 3/2 <4x2 , 3x2 C 6x 9>0 , 3.x C 3/.x 1/>0.This inequalityholdsif x< 3 or x>1

43. jajD a ifandonlyif a 0.Itisfalseif a<0

44. Theequation jx 1jD 1 x holdsif jx 1jD .x 1/, thatis,if x 1 0,or,equivalently,if x 1

45. Thetriangleinequality jx C yjjxjCjyj impliesthat

jxjjx C yj jyj

Applythisinequalitywith x D a b and y D b toget

ja bjjaj jbj

Similarly, ja bjDjb ajjbj jaj.Since ˇ ˇ

jaj jbjˇ

ˇ is equaltoeither jaj jbj or jbj jaj,dependingonthesizes of a and b,wehave

ja bj

SectionP.2CartesianCoordinatesinthe Plane(page16)

1. From A.0;3/ to B.4;0/, x D 4 0 D 4 and y D 0 3 D 3. jABjD p42 C . 3/2 D 5.

2. From A. 1;2/ to B.4; 10/, x D 4 . 1/ D 5 and y D 10 2 D 12 jABjD p52 C . 12/2 D 13

3. From A.3;2/ to B. 1; 2/, x D 1 3 D 4 and y D 2 2 D 4 jABjD p. 4/2 C . 4/2 D 4p2

4. From A.0:5;3/ to B.2;3/, x D 2 0:5 D 1:5 and y D 3 3 D 0 jABjD 1:5

5. Startingpoint: . 2;3/.Increments x D 4, y D 7 Newpositionis . 2 C 4;3 C . 7//,thatis, .2; 4/.

6. Arrivalpoint: . 2; 2/.Increments x D 5, y D 1 Startingpointwas . 2 . 5/; 2 1/,thatis, .3; 3/

7. x2 C y2 D 1 representsacircleofradius1centredatthe origin.

8. x2 C y2 D 2 representsacircleofradius p2 centredat the origin.

9. x2 C y2 1 representspointsinsideandonthecircleof radius1centredattheorigin.

10. x2 C y2 D 0 representstheorigin.

11. y x2 representsallpointslyingonorabovethe parabola y D x2

12. y<x2 representsallpointslyingbelowtheparabola y D x2

13. Theverticallinethrough . 2;5=3/ is x D 2;thehorizontallinethroughthatpointis y D 5=3.

14. Theverticallinethrough .p2; 1:3/ is x Dp2;the horizontallinethroughthatpointis y D 1:3

15. Linethrough . 1;1/ withslope m D 1 is y D 1C1.xC1/, or y D x C 2

16. Linethrough . 2;2/ withslope m D 1=2 is y D 2 C .1=2/.x C 2/,or x 2y D 6

17. Linethrough .0;b/ withslope m D 2 is y D b C 2x

18. Linethrough .a;0/ withslope m D 2 is y D 0 2.x a/, or y D 2a 2x

19. At x D 2,theheightoftheline 2x C 3y D 6 is y D .6 4/=3 D 2=3.Thus .2;1/ liesabovetheline.

20. At x D 3,theheightoftheline x 4y D 7 is y D .3 7/=4 D 1.Thus .3; 1/ liesontheline.

21. Thelinethrough .0;0/ and .2;3/ hasslope

m D .3 0/=.2 0/ D 3=2 andequation y D .3=2/x or 3x 2y D 0

22. Thelinethrough . 2;1/ and .2; 2/ hasslope

m D . 2 1/=.2 C 2/ D 3=4 andequation y D 1 .3=4/.x C 2/ or 3x C 4y D 2

23. Thelinethrough .4;1/ and . 2;3/ hasslope

m D .3 1/=. 2 4/ D 1=3 andequation y D 1 1 3 .x 4/ or x C 3y D 7

24. Thelinethrough . 2;0/ and .0;2/ hasslope

m D .2 0/=.0 C 2/ D 1 andequation y D 2 C x

25. If m D 2 and b Dp2,thenthelinehasequation y D 2x Cp2

26. If m D 1=2 and b D 3,thenthelinehasequation y D .1=2/x 3,or x C 2y D 6

27. 3x C 4y D 12 has x-intercept a D 12=3 D 4 and yintercept b D 12=4 D 3.Itsslopeis b=a D 3=4 y x 3x C 4y D 12

Fig.P.2-27

28. x C 2y D 4 has x-intercept a D 4 and y-intercept b D 4=2 D 2.Itsslopeis b=a D 2=. 4/ D 1=2 y x x C 2y D 4

Fig.P.2-28

29. p2x p3y D 2 has x-intercept a D 2=p2 Dp2 and y-intercept b D 2=p3.Itsslopeis b=a D 2=p6 D p2=3 y x p2x p3y D 2

30. 1:5x 2y D 3 has x-intercept a D 3=1:5 D 2 and y-intercept b D 3=. 2/ D 3=2.Itsslopeis b=a D 3=4 y x 1:5x 2y D 3

Fig. P.2-29

Fig.P.2-30

31. linethrough .2;1/ parallelto y D x C 2 is y D x 1;line perpendicularto y D x C 2 is y D x C 3

32. linethrough . 2;2/ parallelto 2xCy D 4 is 2xCy D 2; lineperpendicularto 2x C y D 4 is x 2y D 6

33. Wehave 3x C 4y D 6 2x 3y D 13 ÷ 6x C 8y D 12 6x 9y D 39:

Subtractingtheseequationsgives 17y D 51,so y D 3 and x D .13 9/=2 D 2.Theintersectionpointis .2; 3/

34. Wehave 2x C y D 8 5x 7y D 1 ÷ 14x C 7y D 56 5x 7y D 1:

Addingtheseequationsgives 19x D 57,so x D 3 and y D 8 2x D 2.Theintersectionpointis .3;2/.

35. If a ¤ 0 and b ¤ 0,then .x=a/ C .y=b/ D 1 represents astraightlinethatisneitherhorizontalnorvertical,and doesnotpassthroughtheorigin.Putting y D 0 weget x=a D 1,sothe x-interceptofthislineis x D a;putting x D 0 gives y=b D 1,sothe y-interceptis y D b.

36. Theline .x=2/ .y=3/ D 1 has x-intercept a D 2,and y-intercept b D 3 y x 3 x 2 y 3 D 1 2

Fig. P.2-36

37. Thelinethrough .2;1/ and .3; 1/ hasslope m D . 1 1/=.3 2/ D 2 andequation y D 1 2.x 2/ D 5 2x.Its y-interceptis5.

38. Thelinethrough . 2;5/ and .k;1/ has x-intercept3,so alsopassesthrough .3;0/.Itsslope m satisﬁes

1 0 k 3 D m D 0 5 3 C 2 D 1:

Thus k 3 D 1,andso k D 2

39. C D Ax C B.If C D 5;000 when x D 10;000 and C D 6;000 when x D 15;000,then

10;000A C B D 5;000 15;000A C B D 6;000

Subtractingtheseequationsgives 5;000A D 1;000,so A D 1=5.Fromtheﬁrstequation, 2;000 C B D 5;000, so B D 3;000.Thecostofprinting100,000pamphletsis $100;000=5 C 3;000 D $23;000

40. 40ı and 40ı isthesametemperatureonboththe FahrenheitandCelsiusscales.

F-50-40-30-20-101020304050607080 . 40; 40/ C D 5 9 .F 32/ C D F Fig.P.2-40

41. A D .2;1/;B D .6;4/;C D .5; 3/

jABjD p.6 2/2 C .4 1/2 Dp25 D 5

jAC jD p.5 2/2 C . 3 1/2 Dp25 D 5

jBC jD p.6 5/2 C .4 C 3/2 Dp50 D 5p2: Since jABjDjAC j,triangle ABC isisosceles.

42. A D .0;0/;B D .1; p3/;C D .2;0/

jABjD q.1 0/2 C .p3 0/2 Dp4 D 2

jAC jD p.2 0/2 C .0 0/2 Dp4 D 2

jBC jD q.2 1/2 C .0 p3/2 Dp4 D 2:

Since jABjDjAC jDjBC j,triangle ABC isequilateral.

43. A D .2; 1/;B D .1;3/;C D . 3;2/

jABjD p.1 2/2 C .3 C 1/2 Dp17

jAC jD p. 3 2/2 C .2 C 1/2 Dp34 Dp2p17

jBC jD p. 3 1/2 C .2 3/2 Dp17:

Since jABjDjBC j and jAC jDp2jABj,triangle ABC isanisoscelesright-angledtrianglewithrightangleat B.Thus ABCD isasquareif D isdisplacedfrom C bythesameamount A isfrom B,thatis,byincrements x D 2 1 D 1 and y D 1 3 D 4.Thus D D . 3 C 1;2 C . 4// D . 2; 2/

44. If M D .xm ;ym / isthemidpointof P1P2,thenthedisplacementof M from P1 equalsthedisplacementof P2 from M : xm x1 D x

45. If Q D .xq ;yq / isthepointon P1P2 thatistwothirdsof thewayfrom P1 to P2,thenthedisplacementof Q from P1 equalstwicethedisplacementof P2 from Q:

xq x1 D 2.x2 xq /;yq y1 D 2.y2 yq

46. Letthecoordinatesof P be .x;0/ andthoseof Q be .X; 2X/.Ifthemidpointof PQ is .2;1/,then .x C X/=2 D 2;.0 2X/=2 D 1:

Thesecondequationimpliesthat X D 1,andthesecond thenimpliesthat x D 5.Thus P is .5;0/.

47. p.x 2/2 C y2 D 4 saysthatthedistanceof .x;y/ from .2;0/ is4,sotheequationrepresentsacircleofradius4 centredat .2;0/

48. p.x 2/2 C y2 D px2 C .y 2/2 saysthat .x;y/ is equidistantfrom .2;0/ and .0;2/.Thus .x;y/ mustlieon thelinethatistherightbisectorofthelinefrom .2;0/ to .0;2/.Asimplerequationforthislineis x D y

49. Theline 2x C ky D 3 hasslope m D 2=k.Thislineis perpendicularto 4x C y D 1,whichhasslope 4,provided m D 1=4,thatis,provided k D 8.Thelineisparallelto 4x C y D 1 if m D 4,thatis,if k D 1=2.

50. Foranyvalueof k,thecoordinatesofthepointofintersectionof x C 2y D 3 and 2x 3y D 1 willalsosatisfy theequation

.x C 2y 3/ C k.2x 3y C 1/ D 0

becausetheycausebothexpressionsinparenthesestobe 0.Theequationaboveislinearin x and y,andsorepresentsastraightlineforanychoiceof k.Thislinewill passthrough .1;2/ provided 1 C 4 3 C k.2 6 C 1/ D 0, thatis,if k D 2=3.Therefore,thelinethroughthepoint ofintersectionofthetwogivenlinesandthroughthepoint .1;2/ hasequation

x C 2y 3 C 2 3 .2x 3y C 1/ D 0; or,onsimpliﬁcation, x D 1.

SectionP.3GraphsofQuadraticEquations (page22)

1. x 2 C y 2 D 16

2. x2 C .y 2/2 D 4,or x2 C y2 4y D 0

3. .x C 2/2 C y2 D 9,or x2 C y2 C 4y D 5

4. .x 3/2 C .y C 4/2 D 25,or x2 C y2 6x C 8y D 0

5. x 2 C y 2 2x D 3 x 2 2x C 1 C y 2 D 4 .x 1/2 C y 2 D 4 centre: .1;0/;radius 2

6. x 2 C y 2 C 4y D 0 x 2 C y 2 C 4y C 4 D 4 x 2 C .y C 2/2 D 4 centre: .0; 2/;radius 2

7. x 2 C y 2 2x C 4y D 4 x 2 2x C 1 C y 2 C 4y C 4 D 9 .x 1/2 C .y C 2/2 D 9 centre: .1; 2/;radius 3.

8. x 2 C y 2 2x y C 1 D 0 x 2 2x C 1 C y 2 y C 1 4 D 1 4 .x 1/2 C y 1 2 2 D 1 4 centre: .1;1=2/;radius 1=2

9. x2 C y2 >1 representsallpointslyingoutsidethecircle ofradius1centredattheorigin.

10. x2 C y2 <4 representstheopendiskconsistingofall pointslyinginsidethecircleofradius2centredatthe origin.

11. .x C 1/2 C y2 4 representsthecloseddiskconsistingof allpointslyinginsideoronthecircleofradius2centred atthepoint . 1;0/

12. x2 C .y 2/2 4 representsthecloseddiskconsistingof allpointslyinginsideoronthecircleofradius2centred atthepoint .0;2/

13. Together, x2 C y2 >1 and x2 C y2 <4 representannulus (washer-shapedregion)consistingofallpointsthatare outsidethecircleofradius1centredattheoriginand insidethecircleofradius2centredattheorigin.

14. Together, x2 C y2 4 and .x C 2/2 C y2 4 representthe regionconsistingofallpointsthatareinsideoronboth thecircleofradius2centredattheoriginandthecircle ofradius2centredat . 2;0/.

15. Together, x2 Cy2 <2x and x2 Cy2 <2y (or,equivalently, .x 1/2 C y2 <1 and x2 C .y 1/2 <1)represent theregionconsistingofallpointsthatareinsideboththe circleofradius1centredat .1;0/ andthecircleofradius 1centredat .0;1/

16. x2 C y2 4x C 2y>4 canberewritten .x 2/2 C .y C 1/2 >9.Thisequation,takentogether with x C y>1,representsallpointsthatliebothoutside thecircleofradius3centredat .2; 1/ andabovetheline x C y D 1.

17. Theinteriorofthecirclewithcentre . 1;2/ andradius p6 isgivenby .x C 1/2 C .y 2/2 <6,or x2 C y2 C 2x 4y<1

18. Theexteriorofthecirclewithcentre .2; 3/ andradius 4 isgivenby .x 2/2 C .y C 3/2 >16,or x2 C y2 4x C 6y>3

19. x 2 C y 2 <2;x 1

20. x 2 C y 2 >4;.x 1/2 C .y 3/2 <10

21. Theparabolawithfocus .0;4/ anddirectrix y D 4 has equation x2 D 16y

22. Theparabolawithfocus .0; 1=2/ anddirectrix y D 1=2 hasequation x2 D 2y

23. Theparabolawithfocus .2;0/ anddirectrix x D 2 has equation y2 D 8x

24. Theparabolawithfocus . 1;0/ anddirectrix x D 1 has equation y2 D 4x

25. y D x2=2 hasfocus .0;1=2/ anddirectrix y D 1=2. y x .0;1=2/ yD 1=2 yDx 2 =2

Fig.P.3-25

26. y D x2 hasfocus .0; 1=4/ anddirectrix y D 1=4

y x yD1=4 .0; 1=4/ yD x 2

Fig.P.3-26

27. x D y2=4 hasfocus . 1;0/ anddirectrix x D 1 y x xD1 . 1;0/ xD y 2 =4

Fig.P.3-27

Version(c)

Version(a)

Version(b)

Fig.P.3-29

a)hasequation y D x2 3

28. x D y2=16 hasfocus .4;0/ anddirectrix x D 4 y x .4;0/ xDy 2 =16 xD 4

Fig.P.3-28

Version(d)

b)hasequation y D .x 4/2 or y D x2 8x C 16

c)hasequation y D .x 3/2 C 3 or y D x2 6x C 12.

d)hasequation y D .x 4/2 2,or y D x2 8x C 14

30. a)If y D mx isshiftedtotherightbyamount x1,the equation y D m.x x1/ results.If .a;b/ satisﬁesthis equation,then b D m.a x1/,andso x1 D a .b=m/ Thustheshiftedequationis

y D m.x a C .b=m// D m.x a/ C b

b)If y D mx isshiftedverticallybyamount y1, theequation y D mx C y1 results.If .a;b/ satisﬁesthisequation,then b D ma C y1,and so y1 D b ma.Thustheshiftedequationis y D mx C b ma D m.x a/ C b,thesame equationobtainedinpart(a).

31. y D p.x=3/ C 1

32. 4y Dpx C 1

33. y D p.3x=2/ C 1

34. .y=2/ Dp4x C 1

35. y D 1 x2 shifteddown1,left1gives y D .x C 1/2

36. x2 C y2 D 5 shiftedup2,left4gives .x C 4/2 C .y 2/2 D 5

37. y D .x 1/2 1 shifteddown1,right1gives y D .x 2/2 2

29. y x .3;3/ 4 .4; 2/ 3 y D x2

38. y Dpx shifteddown2,left4gives y Dpx C 4 2

39. y D x2 C 3, y D 3x C 1.Subtractingtheseequations gives x2 3x C 2 D 0,or .x 1/.x 2/ D 0.Thus x D 1 or x D 2.Thecorrespondingvaluesof y are 4 and 7.The intersectionpointsare .1;4/ and .2;7/

40. y D x2 6, y D 4x x2 .Subtractingtheseequations gives 2x2 4x 6 D 0,or 2.x 3/.x C 1/ D 0.Thus x D 3 or x D 1.Thecorrespondingvaluesof y are 3 and 5 Theintersectionpointsare .3;3/ and . 1; 5/

41. x2 C y2 D 25, 3x C 4y D 0.Thesecondequationsays that y D 3x=4.Substitutingthisintothefirstequation gives 25x2=16 D 25,so x D˙4.If x D 4,thenthe secondequationgives y D 3;if x D 4,then y D 3. Theintersectionpointsare .4; 3/ and . 4;3/.Notethat havingfoundvaluesfor x,wesubstitutedthemintothe linearequationratherthanthequadraticequationtofind thecorrespondingvaluesof y.Hadwesubstitutedinto thequadraticequationwewouldhavegotmoresolutions (fourpointsinall),buttwoofthemwouldhavefailedto satisfy 3x C 4y D 12.Whensolvingsystemsofnonlinear equationsyoushouldalwaysverifythatthesolutionsyou finddosatisfythegivenequations.

42. 2x2 C 2y2 D 5, xy D 1.Thesecondequationsaysthat y D 1=x.Substitutingthisintotheﬁrstequationgives 2x2 C .2=x2/ D 5,or 2x4 5x2 C 2 D 0.Thisequation factorsto .2x2 1/.x2 2/ D 0,soitssolutionsare x D˙1=p2 and x D˙p2.Thecorrespondingvalues of y aregivenby y D 1=x.Therefore,theintersection pointsare .1=p2; p2/, . 1=p2; p2/, .p2;1=p2/,and . p2; 1=p2/

43. .x2=4/ C y2 D 1 isanellipsewithmajoraxisbetween . 2;0/ and .2;0/ andminoraxisbetween .0; 1/ and .0;1/ y x x 2 4 Cy 2 D1

44. 9x2 C 16y2 D 144 isanellipsewithmajoraxisbetween . 4;0/ and .4;0/ andminoraxisbetween .0; 3/ and .0;3/ y x 9x 2 C16y 2 D144

Fig. P.3-43

Fig.P.3-44

45. .x 3/2 9 C .y C 2/2 4 D 1 isan ellipsewithcentreat

.3; 2/,majoraxisbetween .0; 2/ and .6; 2/ andminor axisbetween .3; 4/ and .3;0/ y x .3; 2/ .x 3/2 9 C .yC2/2 4 D1

Fig. P.3-45

46. .x 1/2 C .y C 1/2 4 D 4 isan ellipsewithcentreat

.1; 1/,majoraxisbetween .1; 5/ and .1;3/ andminor axisbetween . 1; 1/ and .3; 1/ y x .x 1/2 C .yC1/2 4 D4 .1; 1/

Fig.P.3-46

47. .x2=4/ y2 D 1 isahyperbolawithcentreatthe originandpassingthrough .˙2;0/.Itsasymptotesare y D˙x=2

y x x 2 4 y 2 D1 yD x=2 yDx=2

Fig.P.3-47

48. x2 y2 D 1 isarectangularhyperbolawithcentreat theoriginandpassingthrough .0; ˙1/.Itsasymptotesare y D˙x. y x x 2 y 2 D 1 yD x yDx

Fig.P.3-48

49. xy D 4 isarectangularhyperbolawithcentreatthe originandpassingthrough .2; 2/ and . 2;2/.Itsasymptotesarethecoordinateaxes. y x xyD 4

Fig.P.3-49

50. .x 1/.y C 2/ D 1 isarectangularhyperbolawithcentre at .1; 2/ andpassingthrough .2; 1/ and .0; 3/.Its asymptotesare x D 1 and y D 2

Fig.P.3-50

51. a)Replacing x with x replacesagraphwithitsreﬂectionacrossthe y-axis.

b)Replacing y with y replacesagraphwithitsreﬂectionacrossthe x-axis.

52. Replacing x with x and y with y reﬂectsthegraphin bothaxes.Thisisequivalenttorotatingthegraph 180ı abouttheorigin.

53. jxjCjyjD 1

Intheﬁrstquadranttheequationis x C y D 1

Inthesecondquadranttheequationis x C y D 1

Inthethirdquadranttheequationis x y D 1 Inthefourthquadranttheequationis x y D 1 y x 1 jxjCjyjD 1 1 1 1

Fig.P.3-53

SectionP.4FunctionsandTheirGraphs (page32)

1. f.x/ D 1 C x2;domain R,range Œ1; 1/

2. f.x/ D 1 px;domain Œ0; 1/,range . 1;1

3. G.x/ Dp8 2x;domain . 1; 4,range Œ0; 1/

4. F.x/ D 1=.x 1/;domain . 1;1/ [ .1; 1/,range . 1;0/ [ .0; 1/

5. h.t/ D t p2 t ;domain . 1;2/,range R.(Theequation y D h.t/ canbesquaredandrewrittenas t 2 C y2t 2y2 D 0,aquadraticequationin t havingreal solutionsforeveryrealvalueof y.Thustherangeof h containsallrealnumbers.)

6. g.x/ D 1 1 px 2 ;domain Œ2;3/ [ .3; 1/,range . 1;0/ [ .0; 1/.Theequation y D g.x/ canbesolved for x D 2 .1 .1=y//2 sohasarealsolutionprovided y ¤ 0

7. y x y x y x y x graph(i) graph(iii) graph(iv) graph(ii)

Fig.P.4-7

Graph(ii)isthegraphofafunctionbecauseverticallines canmeetthegraphonlyonce.Graphs(i),(iii),and(iv) donothavethisproperty,soarenotgraphsoffunctions.

8. y x y x y x y x graph(a) graph(b) graph(d) graph(c)

Fig.P.4-8

a)isthegraphof x.1 x/2,whichispositivefor x>0.

b)isthegraphof x2 x3 D x2.1 x/,whichispositive if x<1.

c)isthegraphof x x4,whichispositiveif 0<x<1 andbehaveslike x near0.

d)isthegraphof x3 x4,whichispositiveif 0<x<1 andbehaveslike x3 near0.

10.

xf.x/ D x4

˙0:50:0625

˙11

˙1:55:0625 ˙216 y x y D x4

Fig. P.4-9

xf.x/ D x2=3

˙0:50:62996

˙11

˙1:51:3104

˙21:5874 y x y D x2=3

Fig.P.4-10

11. f.x/ D x2 C 1 iseven: f. x/ D f.x/

12. f.x/ D x3 C x isodd: f. x/ D f.x/

13. f.x/ D x x2 1 isodd: f. x/ D f.x/

14. f.x/ D 1 x2 1 iseven: f. x/ D f.x/

15. f.x/ D 1 x 2 isoddabout .2;0/: f.2 x/ D f.2 C x/

16. f.x/ D 1 x C 4 isoddabout . 4;0/: f. 4 x/ D f. 4 C x/

17. f.x/ D x2 6x isevenabout x D 3: f.3 x/ D f.3 C x/

18. f.x/ D x3 2 isoddabout .0; 2/: f. x/ C 2 D .f.x/ C 2/

19. f.x/ Djx3jDjxj3 iseven: f. x/ D f.x/

20. f.x/ Djx C 1j isevenabout x D 1: f. 1 x/ D f. 1 C x/

21. f.x/ Dp2x hasnosymmetry.

22. f .x/ D p.x 1/2 isevenabout x D 1: f.1 x/ D f.1 C x/

23. y x yD x 2

y x yD jxj 32. y x yDjxj 1

y x yDjx 2j 34. y x yD1Cjx 2j 35. y x yD 2 xC2 xD 2

y x xD2 yD 1 2 x

y x yD x xC1 xD 1 yD1 38. y x xD1 yD 1 yD x 1 x 39. y x yDf.x/C2 .1;3/ 2 .2;2/ y x yDf.x/ .1;1/ 2

Fig.P.4.39(a)Fig.P.4.39(b)

40. y x yDf.x/C2 .1;3/ 2 .2;2/ y x 1 yDf.x/ 1 .2; 1/ 1

Fig.P.4.40(a)Fig.P.4.40(b)

41. y x yDf.xC2/ . 1;1/ 2

42. y x .2;1/ 13 yDf.x 1/ 43. y x 2 yD f.x/ .1; 1/

44. y x yDf. x/ . 1;1/ 2

45. y x .3;1/ 24 yDf.4 x/

46. y x .1;1/ yD1 f.1 x/ . 1;1/

47. Rangeisapproximately Œ 0:18;0:68 y

x -5-4-3-2-11234 y D

Fig.P.4-47

48. Rangeisapproximately . 1;0:1 [ Œ2:9; 1/ y

x -4-3-2-1123456 y D x 2 x.x C 2/

Fig.P.4-48

49. y -1 1 2 3 4 5 x -5-4-3-2-11234 y D x4 6x3 C 9x2 1

Fig.P.4-49

Apparentsymmetryabout x D 1:5 Thiscanbeconﬁrmedbycalculating f.3 x/,whichturns outtobeequalto f.x/

50. y

1 2 x -5-4-3-2-11234 y D 3 2x C x2 2 2x C x2

Fig.P.4-50

Apparentsymmetryabout x D 1

Thiscanbeconﬁrmedbycalculating f.2 x/,whichturns outtobeequalto f.x/ 51.

Apparentsymmetryabout . 2;2/. Thiscanbeconﬁrmedbycalculatingshiftingthegraph rightby2(replace x with x 2)andthendown2(subtract 2).Theresultis 5x=.1 C x2 /,whichisodd.

53. If f isbothevenandoddthe f.x/ D f. x/ D f.x/, so f.x/ D 0 identically.

SectionP.5CombiningFunctionstoMake NewFunctions(page38)

1. f.x/ D x, g.x/ Dpx 1.

D.f/ D R, D.g/ D Œ1; 1/

D.f C g/ D D.f g/ D D.fg/ D D.g=f/ D Œ1; 1/,

D.f=g/ D .1; 1/. .f C g/.x/ D x Cpx 1

.f g/.x/ D x px 1

.fg/.x/ D xpx 1

.f=g/.x/ D x=px 1

.g=f /.x/ D .p1 x/=x

2. f.x/ Dp1 x, g.x/ Dp1 C x

D.f/ D . 1; 1, D.g/ D Œ 1; 1/.

D.f C g/ D D.f g/ D D.fg/ D Œ 1;1,

52.

Fig.P.4-51

Apparentsymmetryabout .2;1/,andaboutthelines y D x 1 and y D 3 x

Thesecanbeconﬁrmedbynotingthat f.x/ D 1 C 1 x 2 , sothegraphisthatof 1=x shiftedright2unitsandup one.

D.f=g/ D . 1;1, D.g=f/ D Œ 1;1/ .f C g/.x/ Dp1 x Cp1 C x

.f g/.x/ Dp1 x p1 C x

.fg/.x/ Dp1 x2

.f=g/.x/ D p.1 x/=.1 C x/

.g=f/.x/ D p.1 C x/=.1 x/

3. y D x y D x2

Fig.P.4-52

4. y -2 -1 1 x -2 -1 1 y D x y D x3 y D x3 x

5. y x y D x Cjxj y Djxj y D x Djxj y D x

6. y -1 1 2 3 4 x -2-112345 y Djxj y Djx 2j y DjxjCjx 2j

7. f.x/ D x C 5, g.x/ D x2 3

f ı g.0/ D f. 3/ D 2;g.f.0// D g.5/ D 22

f.g.x// D f.x2 3/ D x 2 C 2

g ı f.x/ D g.f.x// D g.x C 5/ D .x C 5/2 3

f ı f. 5/ D f.0/ D 5;g.g.2// D g.1/ D 2

f.f.x// D f.x C 5/ D x C 10

g ı g.x/ D g.g.x// D .x2 3/2 3

8. f.x/ D 2=x, g.x/ D x=.1 x/.

f ı f.x/ D 2=.2=x/ D xI D.f ı f/ Dfx W x ¤ 0g

f ı g.x/ D 2=.x=.1 x// D 2.1 x/=xI

D.f ı g/ Dfx W x ¤ 0;1g

g ı f.x/ D .2=x/=.1 .2=x// D 2=.x 2/I

D.g ı f/ Dfx W x ¤ 0;2g

g ı g.x/ D .x=.1 x//=.1 .x=.1 x/// D x=.1 2x/I

D.g ı g/ Dfx W x ¤ 1=2;1g

9. f.x/ D 1=.1 x/, g.x/ Dpx 1.

f ı f.x/ D 1=.1 .1=.1 x/// D .x 1/=xI

D.f ı f/ Dfx W x ¤ 0;1g

f ı g.x/ D 1=.1 px 1/I

D.f ı g/ Dfx W x 1;x ¤ 2g

g ı f.x/ D p.1=.1 x// 1 D px=.1 x/I

D.g ı f/ D Œ0;1/

g ı g.x/ D qpx 1 1I D.g ı g/ D Œ2; 1/

10. f.x/ D .x C 1/=.x 1/ D 1 C 2=.x 1/, g.x/ D sgn .x/.

f ı f.x/ D 1 C 2=.1 C .2=.x 1/ 1// D xI

D.f ı f/ Dfx W x ¤ 1g

f ı g.x/ D sgn x C 1 sgn x 1 D 0I D.f ı g/ D . 1; 0/

g ı f.x/ D sgn x C 1 x 1 D n 1 if x< 1 or x >1 1 if 1<x<1 I

D.g ı f/ Dfx W x ¤ 1;1g

g ı g.x/ D sgn .sgn .x// D sgn .x/I D.g ı g/ Dfx W x ¤ 0g

f.x/ g.x/f ı g.x/

11. x2 x C 1.x C 1/2

12. x 4x C 4 x

13. px x2 jxj

14. 2x3 C 3 x1=3 2x C 3

15. .x C 1/=x1=.x 1/x

16. 1=.x C 1/2 x 11=x2

17. y Dpx

y D 2 Cpx:previousgraphisraised2units.

y D 2 Cp3 C x:previousgraphisshiftendleft3units.

y D 1=.2 Cp3 C x/:previousgraphturnedupsidedown andshrunkvertically.

D 2 Cpx y D 2 Cpx C 3 y D 1=.2 Cpx C 3/

Dpx

Fig.P.5-17

Fig.P.5-18

19.

27. F.x/ D Ax C B

(a) F ı F.x/ D F.x/

) A.Ax C B/ C B D Ax C B ) AŒ.A 1/x C B D 0

Thus,either A D 0 or A D 1 and B D 0

(b) F ı F.x/ D x ) A.Ax C B/ C B D x

) .A2 1/x C .A C 1/B D 0

Thus,either A D 1 or A D 1 and B D 0

28. bxcD 0 for 0 x<1; dxeD 0 for 1 x<0.

29. bxcDdxe forallintegers x

30. d xeD bxc istrueforallreal x;if x D n C y where n isanintegerand 0 y<1,then x D n y,sothat d xeD n and bxcD n

31. y x y D x bxc

32. f .x/ iscalledtheintegerpartof x because jf.x/j isthelargestintegerthatdoesnotexceed x;i.e. jxjDjf.x/jC y,where 0 y<1 y

y D f.x/

Fig.P.5-32

33. If f isevenand g isodd,then: f 2 , g2 , f ı g, g ı f , and f ı f arealleven. fg, f=g, g=f ,and g ı g areodd, and f C g isneitherevennorodd.Herearetwotypical veriﬁcations:

f ı g. x/ D f.g. x// D f. g.x// D f.g.x// D f ı g.x/ .fg/. x/ D f. x/g. x/ D f.x/Œ g.x/ D f.x/g.x/ D .fg/.x/:

Theothersaresimilar.

34. f even , f. x/ D f.x/ f odd , f. x/ D f.x/ f evenandodd ) f.x/ D f.x/ ) 2f.x/ D 0 ) f.x/ D 0

35. a)Let E.x/ D 1 2 Œf.x/ C f. x/.

Then E. x/ D 1 2 Œf. x/ C f .x/ D E.x/.Hence, E.x/ iseven.

Let O.x/ D 1 2 Œf.x/ f. x/

Then O. x/ D 1 2 Œf. x/ f .x/ D O.x/ and O.x/ isodd.

E.x/ C O.x/ D 1 2 Œf.x/ C f. x/ C 1 2 Œf.x/ f. x/ D f.x/:

Hence, f.x/ isthesumofanevenfunctionandan oddfunction.

b)If f.x/ D E1.x/ C O1.x/ where E1 isevenand O1 isodd,then

E1.x/ C O1.x/ D f.x/ D E.x/ C O.x/:

Thus E1.x/ E.x/ D O.x/ O1.x/.Theleftside ofthisequationisanevenfunctionandtherightside isanoddfunction.Hencebothsidesarebotheven andodd,andarethereforeidentically0byExercise 36.Hence E1 D E and O1 D O.Thisshowsthat f canbewritteninonlyonewayasthesumofaneven functionandanoddfunction.

SectionP.6PolynomialsandRational Functions(page45)

1. x2 7x C 10 D .x C 5/.x C 2/ Therootsare 5 and 2.

2. x2 3x 10 D .x 5/.x C 2/ Therootsare 5 and 2

3. If x2 C 2x C 2 D 0,then x D 2 ˙p4 8 2 D 1 ˙ i The rootsare 1 C i and 1 i . x2 C 2x C 2 D .x C 1 i/.x C 1 C i/

4. Ratherthanusethequadraticformulathistime,letus completethesquare. x 2 6x C 13 D x 2 6x C 9 C 4 D .x 3/2 C 22 D .x 3 2i/.x 3 C 2i/:

Therootsare 3 C 2i and 3 2i

5. 16x4 8x2 C 1 D .4x2 1/2 D .2x 1/2.2x C 1/2.There aretwodoubleroots: 1=2 and 1=2

6. x4 C 6x3 C 9x2 D x2.x2 C 6x C 9/ D x2.x C 3/2.There aretwodoubleroots, 0 and 3

7. x3 C 1 D .x C 1/.x2 x C 1/.Onerootis 1.Theother twoarethesolutionsof x2 x C 1 D 0,namely x D 1 ˙p1 4 2 D 1 2 ˙p3 2 i:

Wehave x3 C 1 D .x C 1/ x 1 2 p3 2 i ! x 1 2 C p3 2 i !

8. x4 1 D .x2 1/.x2 C 1/ D .x 1/.x C 1/.x i/.x C i/.

Therootsare 1, 1, i ,and i

9. x6 3x4 C 3x2 1 D .x2 1/3 D .x 1/3.x C 1/3.The rootsare1and 1,eachwithmultiplicity3.

10. x5 x 4 16x C 16 D .x 1/.x4 16/

D .x 1/.x2 4/.x4 C 4/

D .x 1/.x 2/.x C 2/.x 2i/.x C 2i/:

Therootsare1,2, 2, 2i ,and 2i .

11. x5 C x3 C 8x2 C 8 D .x2 C 1/.x3 C 8/ D .x C 2/.x i/.x C i/.x2 2x C 4/

Threeoftheﬁverootsare 2, i and i .Theremainingtwoaresolutionsof x2 2x C 4 D 0,namely x D 2 ˙p4 16 2 D 1 ˙p3i .Wehave

x5Cx3 C8x2C8 D .xC2/.x i/.xCi/.x aCp3i/.x a p3i/:

12. x9 4x7 x6 C 4x4 D x 4.x5 x 2 4x3 C 4/ D x 4.x3 1/.x2 4/ D x 4.x 1/.x 2/.x C 2/.x2 C x C 1/:

Sevenoftheninerootsare:0(withmultiplicity4), 1,2,and 2.Theothertworootsaresolutionsof x2 C x C 1 D 0,namely x D 1 ˙p1 4 2 D 1 2 ˙p3 2 i:

The requiredfactorizationof x9 4x7 x6 C 4x4 is x 4.x 1/.x 2/.xC2/ x 1 2 C p3 2 i ! x 1 2 p3 2 i ! :

13. Thedenominatoris x2 C 2x C 2 D .x C 1/2 C 1 whichis never 0.Thustherationalfunctionisdeﬁnedforallreal numbers.

14. Thedenominatoris x3 x D x.x 1/.x C 1/ whichiszero if x D 0, 1,or 1.Thustherationalfunctionisdeﬁned forallrealnumbersexcept 0, 1,and 1

15. Thedenominatoris x3 C x2 D x2.x C 1/ whichiszero onlyif x D 0 or x D 1.Thustherationalfunctionis deﬁnedforallrealnumbersexcept 0 and 1

16. Thedenominatoris x2 Cx 1,whichisaquadraticpolynomialwhoserootscanbefoundwiththequadraticformula. Theyare x D . 1 ˙p1 C 4/=2.Hencethegivenrational functionisdeﬁnedforallrealnumbersexcept . 1 p5/=2 and . 1 Cp5/=2

17. x3 1 x2 2 D x3 2x C 2x 1 x2 2

D x.x2 2/ C 2x 1 x2 2

D x C 2x 1 x2 2 :

18. x2 x2 C 5x C 3 D x2 C 5x C 3 5x 3 x2 C 5x C 3

D 1 C 5x 3 x2 C 5x C 3

19. x3 x2 C 2x C 3 D x3 C 2x2 C 3x 2x2 3x x2 C 2x C 3 D x.x2 C 2x C 3/ 2x2 3x x2 C 2x C 3

D x 2.x2 C 2x C 3/ 4x 6 C 3x x2 C 2x C 3

D x 2 C x C 6 x2 C 2x C 3 :

20. x4 C x2 x3 C x2 C 1 D x.x3 C x2 C 1/ x3 x C x2 x3 C x2 C 1

D x C .x3 C x2 C 1/ C x2 C 1 x C x2 x3 C x2 C 1

D x 1 C 2x2 x C 1 x3 C x2 C 1 :

21. As inExample6,wewant a4 D 4,so a2 D 2 and a D p2, b D˙p2a D˙2.Thus P.x/ D .x2 2x C 2/.x2 C 2x C 2/

22. FollowingthemethodofExample6,wecalculate

provided a D 1 and b2 D 1 C 2a2 D 1,so b D˙1.Thus P.x/ D .x2 x C 1/.x2 C x C 1/

23. Let P.x/ D an xn C an 1xn 1 CC a1x C a0, where n 1.BytheFactorTheorem, x 1 isafactor of P.x/ ifandonlyif P.1/ D 0,thatis,ifandonlyif an C an 1 CC a1 C a0 D 0

24. Let P.x/ D anxn C an 1xn 1 CC a1x C a0,where n 1.BytheFactorTheorem, x C 1 isafactorof P.x/ ifandonlyif P. 1/ D 0,thatis,ifandonlyif a0 a1 C a2 a3 CC . 1/n an D 0.Thisconditionsays thatthesumofthecoefﬁcientsofevenpowersisequalto thesumofcoefﬁcientsofoddpowers.

25. Let P.x/ D an xn C an 1xn 1 CC a1x C a0 ,where thecoefﬁcients ak , 0 k n areallrealnumbers,so that ak D ak .Usingthefactsaboutconjugatesofsums and productsmentionedinthestatementoftheproblem, weseethatif z D x C iy,where x and y arereal,then

P.z/ D anzn C an 1zn 1 CC a1 z C a0

D anzn C an 1zn 1 CC a1 z C a0

D P.z/:

If z isa rootof P ,then P.z/ D P.z/ D 0 D 0,and z is alsoarootof P

26. Bythepreviousexercise, z D u iv isalsoarootof P .Therefore P.x/ hastwolinearfactors x u iv and x u C iv.Theproductofthesefactorsisthereal quadraticfactor .x u/2 i 2v2 D x2 2ux C u2 C v2 , whichmustalsobeafactorof P.x/

27. Bythepreviousexercise

P.x/ x2 2ux C u2 C v2 D P.x/ .x u iv/.x u C iv/ D Q1.x/;

where Q1,beingaquotientoftwopolynomialswithreal coefﬁcients,mustalsohaverealcoefﬁcients.If z D u C iv isarootof P havingmultiplicity m>1,thenitmustalso bearootof Q1 (ofmultiplicity m 1),andso,therefore, z mustbearootof Q1,as mustbetherealquadratic x2 2ux C u2 C v2.Thus

P.x/ .x2 2ux C u2 C v2/2 D Q1.x/ x2 2ux

where Q2 isapolynomialwithrealcoefﬁcients.Wecan continueinthiswayuntilweget

.x2 2ux C u2 C v2/m D Qm .x/;

where Qm nolongerhas z (or z)asaroot.Thus z and z must havethesamemultiplicityasrootsof P

SectionP.7TheTrigonometricFunctions (page57) 1. cos 3 4 D

sin 7 12 D sin 4 C 3

D sin 4 cos 3 C cos 4 sin 3

D 1 p2 1 2 C 1 p2 p3 2 D 1 Cp3 2p2

5. cos 5 12 D cos 2 3 4

D cos 2 3 cos 4 C sin 2 3 sin 4

D 1 2 1 p2 C p3 2 ! 1 p2

D p3 1 2p2

6. sin 11 12 D sin 12

D sin 3 4

D sin 3 cos 4 cos 3 sin 4

D p3 2 ! 1 p2 1 2 1 p2

D p3 1 2p2

7. cos. C x/ D cos2 . x/

D cos . x/

D cos. x/ D cos x

8. sin.2 x/ D sin x

9. sin 3 2 x D sin x 2

D sin x 2

D sin 2 x

D cos x

10. cos 3 2 C x! D cos 3 2 cos x sin 3 2 sin x

D . 1/. sin x/ D sin x

11. tan x C cot x D sin x cos x C cos x sin x

D sin2 x C cos2 x cos x sin x

D 1 cos x sin x

12. tan x cot x tan x C cot x D sin x cos x cos x sin x sin x cos x C cos x sin x D sin2 x cos2 x cos x sin x ! sin2 x C cos2 x cos x sin x !

D sin2 x cos2 x

13. cos4 x sin4 x D .cos2 x sin2 x/.cos2 x C sin2 x/

D cos2 x sin2 x D cos.2x/

14. .1 cos x/.1 C cos x/ D 1 cos2 x D sin2 x implies 1 cos x sin x D sin x 1 C cos x .Now 1 cos x sin x D 1 cos 2 x 2 sin 2 x 2 D 1 1 2 sin2 x 2 2 sin x 2 cos x 2 D sin x 2 cos x 2 D tan x 2

15. 1 cos x 1 C cos x D 2 sin2 x 2 2 cos2 x 2 D tan2 x 2

16. cos x sin x cos x C sin x D .cos x sin x/2 .cos x C sin x/.cos x sin x/

D cos2 x 2 sin x cos x C sin2 x cos2 x sin2 x

D 1 sin.2x/ cos.2x/

D sec.2x/ tan.2x/

17. sin 3x D sin.2x C x/

D sin 2x cos x C cos 2x sin x

D 2 sin x cos 2 x C sin x.1 2 sin2 x/

D 2 sin x.1 sin2 x/ C sin x 2 sin3 x

D 3 sin x 4 sin3 x

18. cos 3x D cos.2x C x/

D cos 2x cos x sin 2x sin x

D .2 cos2 x 1/ cos x 2 sin2 x cos x

D 2 cos3 x cos x 2.1 cos2 x/ cos x

D 4 cos3 x 3 cos x

Fig.P.7-19

19. cos 2x hasperiod y x 2

20. sin x 2 hasperiod 4 y x

Fig.P.7-20

21. sin x hasperiod2. y x 2 4 3 1 1 1 y D sin.x/

Fig.P.7-21

22. cos x 2 hasperiod4. y x 1 3 5 1 1

Fig. P.7-22

23. y -3 -2 -1 1 2

24. y -1 1 2 x y D 1 C sin x C 4

25. sin x D 3 5 ; 2 <x<

cos x D 4 5 ; tan x D 3 4 x 5 3 4

Fig. P.7-25

26. tan x D 2 where x isin Œ0; 2 .Then

sec2 x D 1 C tan2 x D 1 C 4 D 5.Hence,

sec x Dp5 andcos x D 1 sec x D 1 p5 ,

sin x D tan x cos x D 2 p5 .

27. cos x D 1 3 ; 2 <x<0

sin x D p8 3 D 2 3 p2

tan x D p8 1 D 2p2 x p8 1 3

Fig. P.7-27

28. cos x D 5 13 where x isin h 2 ;i.Hence,

sin x Dp1 cos2 x D r1 25 169 D 12 13 ,

tan x D 12 5 .

29. sin x D 1 2 ;<x < 3 2

cos x D p3 2 tan x D 1 p3 x 2 1 p3

Fig. P.7-29

30. tan x D 1 2 where x isin Œ; 3 2 .Then,

sec2 x D 1 C 1 4 D 5 4 .Hence,

sec x D p5 2 ; cos x D 2 p5 ;

sin x D tan x cos x D 1 p5

31. c D 2; B D 3

a D c cos B D 2 1 2 D 1

b D c sin B D 2 p3 2 Dp3

32. b D 2; B D 3 B a C b A c 2 a D tan B Dp3 ) a D 2 p3 2 c D sin B D p3 2 ) c D 4 p3

33. a D 5; B D 6

b D a tan B D 5 1 p3 D 5 p3

c D pa2 C b2 D r25 C 25 3 D 10 p3

34. sin A D a c ) a D c sin A

35. a b D tan A ) a D b tan A

36. cos B D a c ) a D c cos B

37. b a D tan B ) a D b cot B

38. sin A D a c ) c D a sin A

39. b c D cos A ) c D b sec A

40. sin A D a c

41. sin A D a c D pc2 b2 c

42. sin A D a c D a pa2 C b2

43. a D 4, b D 3, A D 4

sin B D b sin A a D 3 4 1 p2 D 3 4p2

44. Giventhat a D 2;b D 2;c D 3: C b A c B a

Since a2 D b2 C c2 2bc cos A, cos A D a2 b2 c2 2bc

D 4 4 9 2.2/.3/ D 3 4

45. a D 2, b D 3, c D 4

b2 D a2 C c2 2ac cos B

Thuscos B D 4 C 16 9 2 2 4 D 11 16

sin B D r1 112 162 D p256 121 16 D p135 16

46. Giventhat a D 2;b D 3;C D 4

c2 D a2Cb2 2ab cos C D 4C9 2.2/.3/ cos 4 D 13 12 p2

Hence, c D s13 12 p2 2:12479

47. c D 3, A D 4 , B D 3 implies C D 5 12 a sin A D c sin C ) a D 1 p2 3 sin 5 12 a D 3 p2 1 sin 7 12

D 3 p2 2p2 1 Cp3 (by#5)

D 6 1 Cp3

48. Giventhat a D 2;b D 3;C D 35ı.Then

c2 D 4 C 9 2.2/.3/ cos 35ı,hence c 1:78050.

49. a D 4, B D 40ı , C D 70ı

Thus A D 70ı b sin 40ı D 4 sin 70ı so b D 4 sin 40ı sin 70ı D 2:736

50. If a D 1;b Dp2;A D 30ı,then sin B b D sin A a D 1 2 .

Thus sin B D p2 2 D 1 p2 , B D 4 or 3 4 ,and

C D 4 C 6 D 7 12 or C D

Thus,cos C D cos 7 12 D cos 4 C 3 D 1 p3 2p2 or cos C D cos 12 D cos 3 4 D 1 Cp3 2p2

Hence, c 2 D a 2 C b2 2ab cos C D 1 C 2 2p2 cos C D 3 .1 p3/ or3 .1 Cp3/ D 2 Cp3 or2 p3

Hence, c D p2 Cp3 or p2 p3. =6 p2 1 1 C AB 0 B 00

Fig.P.7-50

51. Let h betheheightofthepoleand x bethedistancefrom C tothebaseofthepole.

Then h D x tan 50ı and h D .x C 10/ tan 35ı Thus x tan 50ı D x tan 35ı C 10 tan 35ı so x D 10 tan 35ı tan 50ı tan 35ı h D 10 tan 50ı tan 35ı tan 50ı tan 35ı 16:98

Thepoleisabout16.98metreshigh.

52. Seethefollowingdiagram.Sincetan 40ı D h=a,therefore a D h= tan 40ı.Similarly, b D h= tan 70ı . Since a C b D 2 km,therefore, h tan 40ı C h tan 70ı D 2 h D 2.tan 40ı tan 70ı/ tan 70ı C tan 40ı 1:286 km

Fig.P.7-52

53. Area 4ABC D 1 2 jBC jh D ah 2 D ac sin B 2 D ab sin C 2

Bysymmetry,area 4ABC also D 1 2 bc sin A b C A h

Fig. P.7-53

54. FromExercise53,area D 1 2 ac sin B. ByCosineLaw, cos B D

2ac .Thus,

Hence,Area D

squareunits.Since, s.s

Thus ps.s a/.s b/.s c/ =Areaoftriangle.

CHAPTER1.LIMITSANDCONTINUITY

Section1.1ExamplesofVelocity,Growth Rate,andArea(page63)

1. Averagevelocity= x t D .t C h/2 t 2 h m/s.

2. h Avg.vel.over Œ2; 2 C h 1 5.0000 0:1 4.1000 0:01

3. Guessvelocityis v D 4 m/s at t D 2 s.

4. Averagevelocityon Œ2;2 C h is .2 C h/2 4 .2 C h/ 2 D 4 C 4h C h2 4 h D 4h C h2 h D 4 C h:

As h approaches0thisaveragevelocityapproaches4m/s

5. x

Averagevelocityoverinterval Œ2;3 is .3 3

6. Averagevelocityover Œt;t C h is 3.t C h/2 12.t C h/ C 1 .3t 2 12t C 1/ .t C h/ t D 6th C 3h2 12h h D 6t C 3h 12 m/s

Thisaveragevelocityapproaches 6t 12 m/sas h approaches0.

At t D 1 thevelocityis 6 1 12 D 6 m/s.

At t D 2 thevelocityis 6 2 12 D 0 m/s.

At t D 3 thevelocityis 6 3 12 D 6 m/s.

7. At t D 1 thevelocityis v D 6<0 sotheparticleis movingtotheleft.

At t D 2 thevelocityis v D 0 sotheparticleisstationary. At t D 3 thevelocityis v D 6>0 sotheparticleis movingtotheright.

8. Averagevelocityover Œt k;t C k is 3.t C k/2 12.t C k/ C 1 Œ3.t k/2 12.t k/ C 1 .t C k/ .t k/ D 1 2k 3t 2 C 6tk C 3k2 12t 12k C 1 3t 2 C 6tk

whichisthevelocityattime t fromExercise7.

9. y 1 2 t 1234 5 y D 2 C 1 sin. t/ Fig.1.1-9

At t D 1 theheightis y D 2 ftandtheweightis movingdownward.

10. Averagevelocityover Œ1;1 C h is 2 C 1 sin .1 C h/ 2 C 1

sin. C h/ h

.h/ C

sin.h/ h D sin.h/ h h Avg.vel.on Œ1; 1 C h

1:0000 0

0:1000 -0.983631643

0:0100 -0.999835515

0:0010 -0.999998355

11. Thevelocityat t D 1 isabout v D 1 ft/s.The“ ” indicatesthattheweightismovingdownward.

12. Wesketchedatangentlinetothegraphonpage55in thetextat t D 20.Thelineappearedtopassthrough thepoints .10;0/ and .50;1/.Onday20thebiomassis growingatabout .1 0/=.50 10/ D 0:025 mm2/d.

13. Thecurveissteepest,andthereforethebiomassisgrowing mostrapidly,ataboutday45.

14. a) proﬁt 25 50 75 100 125 150 175 year 20112012201320142015

Fig.1.1-14

b)Averagerateofincreaseinproﬁtsbetween2010and 2012is 174 62 2012 2010 D 112 2 D 56 (thousand$/yr).

c)Drawingatangentlinetothegraphin(a)at t D 2010 andmeasuringitsslope,weﬁndthatthe rateofincreaseofproﬁtsin2010isabout43thousand$/year.

Section1.2LimitsofFunctions(page71)

1. Frominspectingthegraph y x 1 1 1 y D f.x/

Fig.1.2-1

2. Frominspectingthegraph y x 123 1 y D g.x/

Fig.1.2-2 weseethat lim x!1 g.x/ doesnotexist (leftlimitis1,rightlimitis0)

lim x!2 g.x/ D 1; lim x!3 g.x/ D 0:

lim x!4 .x2 4x C 1/ D 42 4.4/ C 1 D 1

8. lim x!2 3.1 x/.2 x/ D 3. 1/.2 2/ D 0

9. lim x!3 x C 3 x C 6 D 3 C 3 3 C 6 D 2 3

10. lim t ! 4 t 2 4 t D . 4/2 4 C 4 D 2

11. lim x!1 x2 1 x C 1 D 12 1 1 C 1 D 0 2 D 0

12. lim x! 1 x2 1 x C 1 D lim x! 1 .x 1/ D 2

13. lim x!3 x2 6x C 9 x2 9 D lim x!3 .x 3/2 .x 3/.x C 3/ D lim x!3 x 3 x C 3 D 0 6 D 0

14. lim x! 2 x2 C 2x x2 4 D lim x! 2 x x 2 D 2 4 D 1 2

15. limh!2 1 4 h2 does notexist;denominatorapproaches0 butnumeratordoesnotapproach0.

16. limh!0 3h C 4h2 h2 h3 D lim h!0 3 C 4h h h2 does notexist;denominatorapproaches0butnumeratordoesnotapproach0.

17. lim x!9 px 3 x 9 D lim x!9 .px 3/.px C 3/ .x 9/.px C 3/ D lim x!9 x 9 .x 9/.px C 3/ D lim x!9 1 px C 3 D 1 6

18. lim h!0 p4 C h 2 h D lim h!0 4 C h 4 h.p4 C h C 2/ D lim h!0 1 p4 C h C 2 D 1 4

19. lim x! .x /2 x D 02 2 D 0

20. lim x! 2 jx 2jDj 4jD 4

21. lim x!0 jx 2j x 2 D j 2j 2 D 1

22. lim x!2 jx 2j x 2 D lim x!2 1; if x>2 1; if x<2. Hence,lim x!2 jx 2j x 2 does notexist.

23. lim t !1 t 2 1

t 2 2t C 1 lim t !1 .t 1/.t C 1/ .t 1/2 D lim t !1 t C 1 t 1 does notexist (denominator ! 0,numerator ! 2.)

24. lim x!2 p4 4x C x2 x 2

D lim x!2 jx 2j x 2 does notexist.

25. lim t !0 t p4 C t p4 t D lim t !0 t.p4 C t Cp4 t/ .4 C t/ .4 t/ D lim t !0 p4 C t Cp4 t 2 D 2

26. lim x!1 x2 1 px C 3 2 D lim x!1 .x 1/.x C 1/.px C 3 C 2/ .x C 3/ 4

D lim x!1 .x C 1/.px C 3 C 2/ D .2/.p4 C 2/ D 8

27. lim t !0 t 2 C 3t .t C 2/2 .t 2/2

D lim t !0 t.t C 3/ t 2 C 4t C 4 .t 2 4t C 4/

D lim t !0 t C 3 8 D 3 8

28. lim s!0 .s C 1/2 .s 1/2 s D lim s!0 4s s D 4

29. lim y!1 y 4py C 3 y2 1

D lim y!1 .py 1/.py 3/ .py 1/.py C 1/.y C 1/ D 2 4 D 1 2

30. lim x! 1 x3 C 1 x C 1

D lim x! 1 .x C 1/.x2 x C 1/ x C 1 D 3

31. lim x!2 x4 16 x3 8

D lim x!2 .x 2/.x C 2/.x2 C 4/ .x 2/.x2 C 2x C 4/

D .4/.8/ 4 C 4 C 4 D 8 3

32. lim x!8 x2=3 4 x1=3 2

D lim x!8 .x1=3 2/.x1=3 C 2/ .x1=3 2/

D lim x!8 .x1=3 C 2/ D 4

33. lim x!2 1 x 2 4 x2 4

D lim x!2 x C 2 4 .x 2/.x C 2/ D lim x!2 1 x C 2 D 1 4

34. lim x!2 1 x 2 1 x2 4

D lim x!2 x C 2 1 .x 2/.x C 2/

D lim x!2 x C 1 .x 2/.x C 2/ doesnotexist.

35. lim x!0 p2 C x2 p2 x2 x2

D lim x!0 .2 C x2/ .2 x2/ x2.p2 C x2 Cp2 x2/

D lim x!0 2x2 x2p2 C x2/ Cp2 x2

D 2 p2 Cp2 D 1 p2

36. lim x!0 j3x 1j j3x C 1j x

D lim x!0 .3x 1/2 .3x C 1/2 x.j3x 1jCj3x C 1j/

D lim x!0 12x x.j3x 1jCj3x C 1j/ D 12 1 C 1 D 6

37. f .x/ D x 2 lim h!0 f.x C h/ f.x/ h D lim h!0 .x C h/2 x2 h

D lim h!0 2hx C h2 h D lim h!0 2x C h D 2x

38. f.x/ D x3 lim h!0 f.x C h/ f.x/ h D lim h!0 .x C h/3 x3 h

D lim h!0 3x2h C 3xh2 C h3 h

D lim h!0 3x2 C 3xh C h2 D 3x2

39. f.x/ D 1=x lim h!0

f.x C h/ f.x/ h D lim h!0 1 x C h 1 x h

D lim h!0 x .x C h/ h.x C h/x

D lim h!0 1 .x C h/x D 1 x2

40. f.x/ D 1=x2

lim h!0

f.x C h/ f.x/ h D lim h!0 1 .x C h/2 1 x2 h

D lim h!0 x2 .x2 C 2xh C h2/ h.x C h/2x2 D lim h!0 2x C

41. f.x/ Dpx

lim h!0 f.x C h/ f.x/ h D lim h!0 px C h px h

D lim h!0 x C h x h.px C h Cpx/

D lim h!0 1 px C h Cpx D 1 2px

42. f.x/ D 1=px

lim h!0

f.x C h/ f.x/ h D lim h!0 1 px C h 1 px h

D lim h!0 px px C h hpxpx C h

D lim h!0 x .x C h/ hpxpx C h.px Cpx C h/

D lim h!0 1 pxpx C h.px Cpx C h/

D 1 2x3=2

43. lim x!=2 sin x D sin =2 D 1

44. lim x!=4 cos x D cos =4 D 1=p2

45. lim x!=3 cos x D cos =3 D 1=2

46. lim x!2=3 sin x D sin 2=3 Dp3=2

47. x.sin x/=x

˙1:00:84147098

˙0:10:99833417

˙0:010:99998333

˙0:0010:99999983

0:00011:00000000

Itappearsthatlim x!0 sin x x D 1

48. x.1 cos x/=x2

˙1:00:45969769

˙0:10:49958347

˙0:010:49999583

˙0:0010:49999996 0:00010:50000000

Itappearsthatlim x!0 1 cos x x2 D 1 2 .

49. lim x!2 p2 x D 0

50. lim x!2C p2 x doesnotexist.

51. lim x! 2 p2 x D 2

52. lim x! 2C p2 x D 2

53. lim x!0 px3 x doesnotexist. .x3 x< 0 if 0<x<1/

54. lim x!0 px3 x D 0

55. lim x!0C px3 x doesnotexist.(See#9.)

56. lim x!0C px2 x4 D 0

57. lim x!a jx aj x2 a2 D lim x!a jx aj .x a/.x C a/ D 1 2a .a ¤ 0/

58. lim x!aC jx aj x2 a2 D lim x!aC x a x2 a2 D 1 2a

59. lim x!2 x2 4 jx C 2j D 0 4 D 0

60. lim x!2C x2 4 jx C 2j D 0 4 D 0

61. f.x/ D 8 < : x 1 if x 1 x2 C 1 if 1<x 0 .x C /2 if x>0 lim x! 1 f.x/ D lim x! 1 x 1 D 1 1 D 2

62. lim x! 1C f.x/ D lim x! 1C x 2 C 1 D 1 C 1 D 2

63. lim x!0C f.x/ D lim x!0C .x C /2 D 2

64. lim x!0 f.x/ D lim x!0 x 2 C 1 D 1

65. Iflim x!4 f.x/ D 2 andlim x!4 g.x/ D 3,then

a)lim x!4g.x/ C 3 D 3 C 3 D 0 b)lim x!4 xf.x/ D 4 2 D 8

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CHAPTER XII

BURIAL RITES

During my stay in Dulukan, Mafel, one of the most popular and respected men of the district, was slowly dying of a malignant cancer of the face, which was destroying his lower jaw and penetrating deep into his throat. Day by day we had reports of his courageous and patient suffering, and of the devotion to him of his only daughter, Gyeiga, who never left his side, doing everything in her power to minister to his needs, trying to give him food, and fanning him night and day to keep the swarms of mosquitoes and flies from annoying him as he lay propped up on his mat spread on the hard floor. He had been treated for some weeks in the government hospital at the other end of the island, but when he found he was gradually becoming worse, he begged to be taken back to his own home where he could see his friends and pass away quietly; he was carried thither and the skill of all the most renowned mach-machs was invoked to dispel the demons of disease and enlist Yalafath’s sympathy and protection in behalf of the patient sufferer. In spite of all their energetic efforts, however, slow starvation reduced him to a mere skeleton, and finally word was brought to us early one morning that poor Mafel’s tafenai had wandered away from him in the night and had gone to Falraman. The devotion of Gyeiga did not cease, even then; she still sat by the side of the repulsive corpse, fanning untiringly, and wailing forth some disjointed snatches of a death song, wherein were recounted the good qualities and kindnesses of him who had been indeed a father to her; the dirge was constantly interrupted with a refrain—O Mafel, O garfuku,—“O Mafel, O poor one!”

A messenger was immediately dispatched to the far-northern end of the island to notify Mafel’s uncle, Livamadai, his nearest relative, an important chief and momok man; on him rested the decision as to whether the body should be buried on the following day, or kept two or three days longer. To defer the funeral is a tribute of honour to the

corpse; haste in burial affords the chance of a visitation from the athegith, wherefrom sickness and mishaps surely follow.

Old Livamadai, toothless, bald, and bent in the knees, hobbled down the next day and decided that the following day, or the third day after death, would be a delay sufficient to show respect to Mafel’s remains. Poor Gyeiga had one more weary night of vigil; they said she never left the side of the body and took barely a mouthful of food or a wink of sleep all those three long days and nights. The atmosphere of the house was truly unbearable; I went to ask her if I might come to the funeral, and if she had any objection to my taking some photographs, and, after expressing my deep sympathy and receiving her willing permission, I retired as quickly as I possibly could from that inexpressibly noisome and dark house of death.

On the following day there was a constant procession passing our house on the way to the funeral; each person bearing a gift for the corpse, usually strings of pearl-shell money or single large shells; some of the wealthy and liberal friends brought a fei of such size that it required two men to carry it.

I went to the house with Fatumak a little after noon; they said that Mafel probably would not be buried until late in the day.

When we arrived at the house I noticed that the space about it, enclosed by a fence of light bamboo, was occupied by women only; Fatumak explained to me that he would have to leave me at the entrance, if I intended to go in; it was against custom for any, except women and the slave class, to enter the yard of a dead man’s house while his body was unburied; of course, I, as a foreigner, would not be restricted.

I set up my camera and focussed it on that side of the house where they would probably break through the walls to bring out the body,—through a doorway it is never carried, it inevitably brings ill luck to the living inmates,—then I rejoined Fatumak just outside the fence to watch and wait and ask questions. It was evident from the number of presents deposited at one end of the yard, beyond the group of guests, that Mafel had been very popular and that his friends were wealthy, and lavish withal, both in money and sympathy.

“Yes,” whispered Fatumak, “Mafel was truly a fine man; we all liked him; those presents will be pretty nearly all returned after he is buried; they bring them to show their sorrow, but it is always expected that they will get them back again.”

FUNERAL GIFTS OF STONE MONEY AND PEARL SHELLS

The women, in groups about the yard, had all brought their work with them, and, sitting cross-legged on the ground, from time to time, in subdued funereal whispers wherein sibilants always seem to predominate, they gossipped with one another and kept their fingers busy, some by plaiting little pouches to hold cigarettes and tobacco, some by repairing their leafy skirts, and others by making new betel baskets; but all were solemn and subdued in the presence of death and sorrow.

It was one of those gorgeous, lazy, tropical days when the very air is idle and a sabbath stillness holds everything; there was not even the hum of an insect or the piping of a seagull to break the

quiet, and only every now and then was there a breath of air strong enough to make the palm leaves rustle softly. Once, the silence was rudely broken by the thud of a ripe coconut falling to the ground, which for a brief period diverted the solemn contemplation of death to thoughts of commerce. A hush brooded over everything, even the irrepressible “tomboy” Kakofel, sat demurely beside a group of women, rolling a store of cigarettes for herself; Fak-Fintuk, Libyan, Gumaon and the other obstreperous boys were, for once, unseen. The presents consisted of six or more good sized fei of fine quality, six or seven baskets full of shell money, and numerous single strings of the same; really quite a fortune. All gifts were deposited with a good deal of display by the donors at one end of the yard in front of the house; for this service they were allowed to enter the yard, but were expected to withdraw as soon as their offering had been deposited and duly appreciated. After an hour or more waiting, five very solemn men of the Pimlingai tribe filed into the yard and sat down quietly in the background; then there was a little stir among the women as they shifted their positions to get a better view of the side of the house whence the funeral procession was to set out, and after a short pause,—for no move must be made suddenly, the Pimlingai brought forward a litter of bamboo poles covered with matting of woven coconut fronds. This they carried into the house and on it they placed the emaciated body of Mafel with his knees drawn up and tied together and his hands folded across his body. The side wall of the house of reeds and matting was taken down and through the opening the litter bearing the corpse brought out and placed upon the ground. Gyeiga’s chant grew louder and louder within the house and was no longer a mere sing-song, but a passionate wail of sorrow, when, accompanied by her two sisters-in-law (I think), she followed the litter out of the house and took her place beside it on the ground. The eyes of all three women were streaming with tears, but Gyeiga was the only one who wept aloud. The Pimlingai again retreated to the background, and Gyeiga, sitting cross-legged beside the corpse, placed two large pearl shells upon his chest, talking to him in a pleading, plaintive voice and looking directly in his horribly disfigured face. The old women in the listening and sympathising crowd, from time to time seemed to mutter an approval of her

sentiments, and the wrinkled, parchment cheeks of many of them were wet with tears. Then she arose and brought two more equally fine shells from the house and placed them on top of the others with another short speech to the corpse. As soon as this was done, the Pimlingai came forward and wrapped the matting completely round the body, leaving only the top of the head bare. Two of them picked up the burden and the third placed a pole on their shoulders and to this tied the sides of the litter so that the weight was distributed evenly between their shoulders and arms. They wheeled around and rapidly walked out through an opening in the bamboo fence back of the house; Gyeiga and her two chief mourners and three or four other women followed, wailing loudly.

GYEIGA PLACING TWO PEARL SHELLS ON HER FATHER’S CORPSE

With Vincenti (Friedlander’s Christianised servant from Guam) I followed after them, barely able to keep up with their rapid pace over the slippery and irregular boulders of stone and coral with which the side paths of the island are paved. The wailing was kept up

continuously by the different members of the party; when one became tired, the next took it up, and so on, until each had wailed in turn, and then Gyeiga began anew.

In and out we wound through jungle paths, now overarched with grey-green bamboos, now hemmed in with hedges of tall, variegated crotons; past small clusters of houses where the people stared to see a funeral party followed by a staggering leather-shod white man and a lad with a queer looking box on a stick over his shoulder. Then down to the flat lands, past the taro patches and plantations of yams, and through a deserted tapal, or village, of small houses used as a maternity-ward,—strange place for a funeral procession to invade. There were no inmates at that time in the little houses except numerous small grey lizards with brilliantly blue tails, that darted in all directions like little electric sparks in the sunshine on the thatched sides of the houses.

We seemed to double on our tracks and zig-zag hither and thither, until at length we passed through a Pimlingai village where three or four more women and eight or ten children of the village silently joined the procession. A short distance beyond this village, the men bearing the litter turned off the path directly into the thick undergrowth, and pushing through after them, we came out into a clearing about a hundred feet in diameter. At one side there were several young coconut palms just sprouting above the ground and scattered here and there were low mounds and piles of mosscovered stones, six or eight perhaps in all, graves of those who had gone before. The Pimlingai put down the litter bearing the last remains of Mafel close to one of these mounds, which appeared to have been very recently made and whereon still rested the decaying remnants of a similar litter; they told me afterward it was the grave of his wife who had died only a few months before.

As soon as the litter rested upon the ground, Gyeiga sat down beside it and tenderly unfastened the matting which covered the body and once more exposed it to view, and with a palm leaf began again her untiring fanning and low wailing, constantly repeating “O Mafel! O my poor one!” The Pimlingai disappeared for a minute or two in the thick jungle and undergrowth, and then emerged with long

poles sharpened at one end with which they proceeded to loosen the ground at the far side of the litter with its half reclining corpse.

The chief mourners who had accompanied Gyeiga set to work plaiting rough baskets or hampers of coconut fronds, and in these the loose earth was gathered up in handfuls by the Pimlingai, and piled to one side or carried off and scattered in the jungle. After making these baskets, the women busied themselves collecting stones and flat pieces of coral rock wherewith to line and to cover the grave.

While this was going on, the women and children, twenty-five or more, who had joined the procession at the last Pimlingai village, sat silently, quite far off at the opposite side of the graveyard; I was trying to get my camera in position so as to get a view of the gravediggers, but the only available spot placed them directly between me and the declining sun, so I was forced to refrain from the attempt. While I was testing my position, I frequently heard the female spectators of the Pimlingai whispering Tokota, Tokota, the name by which I was known to them. It was an attempt at “Doctor,” which they had heard Friedlander call me. Glancing up, I noticed one of the women looking at me and making motions up and down her arm. Then I comprehended that they wanted to see the Japanese tattooing there. I went over to her and, having rolled up my sleeves, received a liberal palming and rubbing; amazed at the various colours, she and the others could not believe they were not mere paint which a vigorous rub, aided by moisture from the tongue, would remove. The sight of a Japanese carp tattooed on the calf of my leg called forth such loud expressions of admiration, that I was afraid I was sadly interfering with the proprieties of the mournful occasion, so I drew up my stockings and hastily retired.

When the grave was dug out about two and a half feet deep, by three feet long, and eighteen inches wide, the Pimlingai lifted Mafel on the mat whereon he rested and placed him in the grave, with his head toward the setting sun. Before putting any earth over him, one of the Pimlingai took, as payment for their labours, two of the pearl shells that had been placed upon the corpse; the other two were buried with him; he must not arrive empty handed in Falraman.

As soon as the body was placed in the grave the wailings of Gyeiga and her chief mourners were redoubled, and over and over again they bade him goodbye and reiterated “O Mafel! O my poor one!” When the grave was nearly filled in, a sprouting coconut was planted at the head and banked round with earth and lumps of coral. It was to provide food for Mafel on his journey to Falraman, and also to furnish oil not only for light, but also for his hair; a coconut is always thus planted at the head of a corpse,—witness the young trees in the graveyard. Slabs of stone and coral were piled up all about the grave for a distance of two feet, and earth tightly packed in the crevices, so that the big lizards,—“monitors,” the only large reptile on the island,—should not disturb the body.

Until the last block of stone and handful of earth was placed on the grave, Gyeiga and the mourners never ceased wailing; but the very minute that all was finished and patted down, they ceased abruptly. Gyeiga wiped away her tears, lit a fresh cigarette and disappeared in the jungle.

It was too dark for photographs, so I packed up my camera and, following Vincenti, I too plunged into the undergrowth, and in an incredibly short time, as it seemed, was in Dulukan. I learned that the route we had followed to the graveyard was as circuitous as could be devised, and that this was always the custom in the burial of people of importance; a poor man is hurried as quickly as possible to his grave, but a wealthy man is taken past as many houses as possible and in a roundabout way, so that the grief of his relatives may be seen and heard far and wide.

On questioning Fatumak after the burial, I found that the manner of death has much to do with the position in which the body is interred; if a man dies of an ordinary disease or of old age, he is buried with his head to the west and his knees drawn up, as in Mafel’s case; if he dies in battle, he is buried with his head to the north and his legs and body are perfectly straight; if he dies of a cough,—consumption,—he is buried with his knees drawn close to his breast, and with his face looking downward. The graves, as a rule, are very modest little mounds in the quiet seclusion of the bush near some Pimlingai village, but when a great chief dies, a large

platform of flat stones, such as the houses are built on, is constructed over the grave, and the departing tafenai is speeded on its way to Falraman with feasting and dancing.

Such is life and death on the happy little island of Uap; at least as I saw it in a two months’ residence; they are delightful people to visit now that Germany exerts a truly paternal care over them and perpetuates their naturally mild temper by strictly prohibiting the introduction of alcohol among them.

When, early one morning, I sailed away from Dulukan in Friedlander’s barge bound for Tomil Bay, to meet the steamer and depart for Sydney, all my friends were on hand to see me off,—Migiul and Lemet, who had contributed to my collection of tattoo marks and cat’s-cradle figures; Lian, who had helped in many ways to get specimens for my collection; Tomak, of the strong voice, who had contributed many a song on the phonograph; Gamiau, who had been foremost in getting up the dance; even Kakofel, whose sensitive feelings I had grievously wounded by calling her “Kakofel Kan,” was there, but she stayed in the background and only stared when I shook her hand for goodbye. Little Pooguroo, my earliest and faithfullest wee friend, stood on the very extremity of the jetty, her little brown body glistening in the warm light of the rising sun, and her large black eyes following me wonderingly as we were gradually poled out into the channel of the lagoon.

Just as we made the first turn and Dulukan had faded from sight, we met good old Fatumak on his “barco;” he shouted to me a few of the auspicious phrases which are used to fisherman as they set out to sea, and I shouted back to him goan e gup! which means “I am going, but I shall return,”—a phrase of courtesy when one leaves a party of friends and expects to return before long,—it about

corresponds to “Auf Wiedersehn.” Indeed the words were uttered in all sincerity. Who would not wish, at least for a season, to renew, “through the verdurous glooms” of the tropics, a life as simple, as equable, as hospitable as that which I received at the hands of the natives of Uap.

UAP GRAMMAR

Only a few days before my departure from Uap, I received through the kindness of Padre Cristobal de Canals, a grammar, written in Spanish, of the language of the island. The small volume of a hundred and forty-four pages bears the following title: Primer Ensayo de Grammatica de la lengua de Yap (Carolina Occidentales) con un Pequeno Diccionario y varias Frases en forma Diálogo. Por un Padre Capuchino, Misionero de aquellas islas. Manila. Imprenta del Collegio de Santo Tomas, á cargo de D. Gervasi, Memije, 1888.

In a short preface the Padre tells us that the modest treatise is the work of a residence in the island of Uap of about a year.

It is almost needless to remark that when a language has never been set forth in writing, its forms and even its pronunciation are as shifting as the sands of a beach. The only object of those to whom it is native is to understand and be understood. Let these two ends be gained, and all the accidents of grammar are superfluous and pronunciation will fall under no critic’s condemnation. That this is true as regards pronunciation, sufficient proof is come under my own observation; in the twenty years that elapsed between the date of the Padre’s grammar and my sojourn in the island, the pronunciation showed marked variation between that recorded in the Grammar and that current in the island when I visited it.

Furthermore, it may be noted, I fear, that the Padre, in certain cases, especially in the conjugations of verbs, failed to observe that what he assumes to be a variation in structure decided enough to constitute a separate conjugation, is, after all, merely a change due to euphony, or due to a colloquial contraction, as we find it in all languages, such as, for instance, we have in English in our familiar haven’t, where, of course, n’t is not a part of the verb.

In these circumstances I have deemed it wisest to set forth the Etymology and Syntax in the briefest and most concise way, and trust to phrases and the vocabulary as supplemental to the mother wit of the traveller in his communications with the simple-minded natives of this truly charming island, and I am bound to add that the

novice will never find there severe critics of grammatical or linguistic blunders.

It is to be borne in mind that the language of Uap belongs, certainly to a large degree, to the Agglutinating Group; and, possibly, the more intimate our familiarity with it, the more distinctly we should recognize as compounds words, which we now regard as simple, and analyze them into their component parts. For instance, the definite article “the” is faré; “those,” fapi; “those two,” fagali. Here fa is evidently a root and the affix pi we know to be the sign of the plural, but the meanings of ré and gali are lost.

There are no grammatical genders, that is, there are no affixes, suffixes, or terminations to indicate genders, but pumawn, man, and pin, woman, follow the noun when sex is to be emphasized. We have the same poverty in English in expressing the gender of certain animals, such as: she-wolf, he-goat, she-bear, etc.

There appears to be no Indefinite Article, and for even a Definite Article there seems to be no very great use. It is as follows, for all genders:

Singular faré the Plural fapi those Dual fagali those two

E: The man faré pumawn; the woman faré pin; the house faré naun; the men fapi pumawn; the women fapi pin; the two women fagali pin; etc.

The second syllable of the plural fapi is also used to express the plural, e.g., the children—pi abetir; the people in a village—pi u binau.

Before going further into the maze of Uap words and their arrangement in sentences I am impressed with the advisability of quoting from Professor Basil Hall Chamberlain’s “A Hand-Book of Colloquial Japanese” (page 11) in order to give an excuse and to ask pardon for giving a comparison and classification of one of these Far Eastern languages in terms used in the grammars of the other side of the world.

“A word as to the parts of speech in Japanese. Strictly speaking there are but two, the verb and the noun. The particles or ‘postpositions’ and suffixes, which take the place of our prepositions, conjunctions, and conjugational terminations, were themselves originally fragments of nouns and verbs The pronoun and numeral are simply nouns The true adjective (including the adverb) is a sort of neuter verb But many words answering to our adjectives and adverbs are nouns in Japanese Altogether our grammatical categories do not fit the Japanese language well They have only been adhered to in this work in so far as they may serve as landmarks familiar to the student.”

PRONOUNS

The are igak—I, igur—thou, tsanem—he, she or it. Igak is thus declined:

Nominative igak I

Genitive and ablative rak of me

Dative gufanei for me

Accusative and dative ngok me; to me

A curious refinement is to be noted in the dual and plural of this first personal pronoun; each possesses two inflections, namely, one conveying the ordinary idea of duality or plurality, such as gadou— we two, and gadad—we; and a second conveying the idea that the present company is alone referred to and that all others are excluded. For instance, gadou u Rul means simply “we two men of Rul,” but should the two men be joined by a third whom they did not wish to be included in the pronoun, the phrase would be gomou u Rul, that is, “we two men, and we two men alone, of Rul.” Thus, also, should a man happen to address the assembled people of his district, he would say: Gadad pi u Rul, i.e., “We the people of Rul,” but if he wished to express the idea that he refers to their own district, to the exclusion of all others, he would say, Gomad pi u Rul.

The two numbers, dual and plural, of the first person, are thus declined:

Dual nominative gadou we two Genitive and ablative rodou of us two, with us two Dative nḡadafanou for us two

Accusative nḡodou us two; to us two

D (Exclusive Form):

Nominative gomou we two only

Genitive and ablative romou of or with us two only

Dative kufanu for us two only

Accusative nḡomou us two only

Nominative gadad we

Genitive and ablative rodad of us; or, with us

Dative nḡadafaned for us

Accusative and dative nḡodad us; or, to us

P (Exclusive):

Nominative gomad we only

Genitive and ablative romad of us; or, with us only

Dative goufaned for us only

Accusative and dative nḡomad us only; or, to us only

The second person is thus declined:

Nominative igur thou

Genitive and ablative rom of thee; or, with thee

Dative mufanei for thee

Accusative and dative nḡom thee; or, to thee

Nominative gumu you two

Genitive and ablative romu of or with you two

Dative mufanu for you two

Accusative and dative nḡomu you two, or to you two

Nominative gumed you

Genitive and ablative romed of you, or with you

Dative mufaned for you

Accusative and dative nḡomed you, or to you

The third person:

Nominative tsanem, fanem he, she, it

Genitive and ablative rok of or with him, her, it

Dative fanei for him, her, it

Accusative and dative nḡak him, her, it, or to him, her, it

Nominative galitsanem they two

Genitive and ablative rorou of, or with the two

Dative rafanou for the two

Accusative and dative nḡorou them two, or to the two

Nominative pitsanem they

Genitive and ablative rorad of them, or with them

Dative rafaned for them

Accusative and dative nḡorad them, or to them

D are of three kinds, namely, personal, impersonal (i.e., relating to things animate and inanimate), and partitive (i.e., relating to parts or pieces of objects). The personal demonstrative pronouns are:

tsanei, or anei this [The abbreviation, anei, apparently, is only for euphony.]

tsanir, or anir that tsanem, or anem that yonder fatsa that far off, unseen, out of sight

galitsanei, or galianei these two galitsanir, or galianir those two galitsanem, or galianem those two yonder

pitsanei, or pianei these pitsanir, or pianir those pitsanem, or yad those yonder

Impersonal demonstrative pronouns, referring to animals and things: binei—this; binir—that; binem—that yonder; tinei—these; tinir —those; tinem—those yonder.

Partitive demonstrative pronouns: kinei—this piece of; kinir—that piece of; kinem—that piece yonder of. Tsikinei (if the piece is very

small); tsikinega (if the piece is very large).

The , when applied to anything which has no relation to our body, is the genitive of the personal pronoun, placed after the noun: purpur rak—my hat; naun rom—thy house; ton rok—his hatchet; mad romad—our clothes; domunemun romed —your food; uelduk rorou—the field of those two.

The possessives of nouns signifying parts of the body, or things relating to or proceeding from it, are formed as follows: the noun loses its last letter, when this is an i, and sometimes the last syllable, when it happens to be ngin, and then the last syllable of the genitive of the personal pronoun is used as a possessive suffix. For the first or second person singular, the suffix is -ak or -ek for the first person, -am or -em for the second person. I cannot, however, detect any rule whereby the vowel should be changed, nor can any rule be given for the third person.

E: lungai mouth; lungak my mouth; lungam thy mouth; lungan his mouth; lunga-dad our mouths; lunga-med your mouths; lunga-rad their mouths; lolugei head; lolugek my head; lolugem thy head, etc

R P.—The idea of relation is expressed by the particle ni. Thus, faré abetir ni ior—the child that cries; nu ni keb rain that falls.

By means of this ni, is formed the interrogative mini, which may be placed either before or after the noun. Thus, mini igur—who art thou? mini e romed—which of you? pianir mini—who are those women?

The following interrogative particles are used for animals and inanimate objects: mang—what? beninḡan—which one? tininḡan— which ones? Galininḡan—which of those two?

When mini precedes a demonstrative personal pronoun, beginning with a consonant, and mang precedes a neuter demonstrative pronoun, they are followed by the particle e. Thus, mini e tsanei—who is this person? mang e binei—what is this (thing or animal)?

The are the following: tareb, or tab—one, the one; be—the other; dari—no one, no thing. Thus, tareb e pumawn ni keb—the man who comes; bine e naun, naun ku bë—this house is the house of the other man; dari pumawn u naun—there is no man at home.

VERBS

There is no substantive verb. The past, present, and future state must be gathered from the drift of the sentence. Its place is filled, however, by two particles ni and e; of which we have already had examples. Thus, faré māāb ni bin—the door is open; matsalabok e naun—clean is the house.

N. B. After the three personal pronouns, these particles are omitted. Thus, igak alid—I am dirty; igur matsalabok—thou art clean; tsanem fel—he is good. They are also omitted in negative sentences when dagathi, not, is used. Thus, dagathi alid—it [is] not dirty; dagathi Tomak—it [is] not Tomak; faré abetir dagathi fel—the child [is] not good; faré gatu dagathi baga—the cat [is] not large. When, however, for the sake of emphasis, the predicate precedes the subject, then these particles are used. Thus, dagathi fel e abetir—it is not a good child; dagathi baga e gatu—it is not a large cat.

“It is,” “there is,” “there are,” are sometimes expressed by kabai. Thus, kabai u nifi—it is in the fire; kabai bȯȯr wu—there are many betel nuts.

When kabai is used in the sense of “to have,” it is followed by the genitive of the possessor. Thus, kabai debdeb rak—I have a box; kabai piri olum rok—he is very cold.

“Not to be,” and “not to have,” when referring to inanimate objects, or to the dead, are expressed by dari Thus, dari e lugud rok —I have no cigarette; dari e morau—there is no [ripe] coconut.

When they refer, however, to animate objects, dari may be also used, but likewise demoi (sing.), darmei (dual), and darmed (plural). Thus, pumawn demoi u mu—the man is not in the canoe; fouap darmei fakam ni fel—yesterday your two children were not good; darmed fapi abetir u naun—the boys are not in the house.

In the little Spanish and Uap Grammar, of which I have already spoken, and to which I wish always to express my obligation, though I have by no means followed it, verbs are divided into six conjugations, and paradigms of all are given. For reasons which are to me sufficient, this division appears too elaborate, and a little arbitrary in dealing with an unwritten language, which varies from generation to generation. Inasmuch as there is no Uap literature and the only object in learning the language is for the purpose of conversation, I think it better, to judge by my own experience, to learn these various inflections from phrases and a vocabulary, rather than to memorize page after page of paradigms. Accordingly, the conjugation of only one verb is here given, merely to show the general inflection,—premising that there are, what we might naturally expect, only three tenses: the present, past and future. Thus, we may conjugate non, to speak, where non is not an infinitive, but merely a root:

P T

gu-non

Singular

Plural absolute

Plural restrictive

Dual absolute

Dual restrictive

I speak mu-non thou speakest be-non he speaks

da-non-ad we all speak

da-non-ed you all speak

da-non-od they all speak

gu-non-ad we alone speak

mu-non-ad you alone speak

ra-non-ad they alone speak

da-non-ou we two speak

gu-non-ou we two alone speak

mu-non-ou you two alone speak

ra-non-ou they two alone speak

P T

Singular kogu-non

I spoke, I have spoken komu-non thou hast spoken i-non, or ke-non he spoke, he has spoken

Plural absolute kada-non-ad, -ed, -od we, you, they, all spoke

Plural restrictive kogu-non-ad we alone spoke

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