ucsd博士一年級課程 mathematics of economists 習題指南

Economics205,Fall2010

QuizI

August27,2010

Instructions.Trytoanswerall3problems.(Readallofthequestionsnow

andstartontheonesthatseemeasiest).Makeyouranswersascompleteand

rigorousaspossible.Wlienyoucomputeaderivativesay“Thisstepfollowsfrom

thechainrule^^or“Becausethederivativeofasumisthesumofthederivatives

...Whenyoutakealimit,invokethenecessaryresults(orgiveadirectproof

withes).

Informalandintuitiveargumentsarebetterthannothing.

1.Let/beadifferentiablefunction.Calculatethederivativeofthefunction

hdefinedineachoftheproblemsbelow.Ifyouneedadditionalassump-

tions,makethemexplicit:

(a)h(x)=log/(x2)

(b)h(x)=/(logx)

(c)/i(x)=elog工

2.Calculatethelimitsindicatedbelow.

(a)lim7Too

(b)lim①T5

(c)lima._>0+xlogx.

3.Letf:[0.1]—>R.

(a)Provethatif

|a-41/2>(a)-/(b)|foralla,6€[0,1],(1)

thenfiscontinuouson(0,1).

(b)Giveanexampleofanon-constantfunctionfthatsatisfies(??).

[Provethatyourexamplesatisfiesthecondition.]

1

Economics205.Fall2010

QuizI,PossibleAnswers

August27,2010

Comments.Scoresoutof100.Range:46-96.Average:81,Median:83.

Allocation:40/40/30.

Firstquestion:OK,butsomepeopledidnotnoterangeofvalidity.Third

question:Thisiswheremostpeoplelostpoints.

Minimal(butpositive!)deductionfornotprovidingjustification.Deduc-

tionsfornotexplainingyouranswerswillincreaseonfutureassessments.

1.Let/beadifferentiablefunction.Calculatethederivativeofthefunction

hdefinedineachoftheproblemsbelow.Ifyouneedadditionalassump-

tions,makethemexplicit:

(a)h(x)=log/(a;2).)bytheChainRule(twice).You

canonlydothiscomputationwhen/(T2)>0.

(b)h(x)=/(logT).九'(①)=/'(log①)//bythechainruleandtherule

fordifferentiatingloga?.Needx>0forhtobedefined.

logx

(c)h(x)=e=xysoh!(x)=1

2.Calculatethelimitsindicatedbelow.

i;m—n

(a)UIURTOO3rl3+6-u

Givene>0,letAT=1/e

n2—1n2—1

0<―-<—<6.

3n3+6n3

(b)“ru;工二十5言=2.3.

(Limitofcontinuousfunctionratioofpolynomialswithnonzero

denominator.Sobycontinuity,justevaluate.)

lima._+0+xlogx=lim￡ToiCanuseL'Hopital'sRuleonlast

expressiontoobtain:

limlim等1而4=0

Z

6T0+CT0+l/xX->0+—1/X

3.Letf:[0?1]tR.

1

(a)Provethatif

1/2

\a-b\>|/(Q)-/(d)|foralla,be[0,1],(1)

thenfiscontinuouson(0,1).

Theinequalitysaysthat-f(a)-f(b)-is“sandwiched”betweenthe

constantfunctionequaltozeroandthefunctiong(x)=\a—①?

Sincegiscontinuousata(acceptedwisdom),linib->a1/(。)一/(6)|=

0.Thisiswhatweneededtoshow.

(b)Giveanexampleofanon-constantfunctionfthatsatisfies(1).

[Provethatyourexamplesatisfiesthecondition.]

Therearemanypossibilities.f(x)=xworksbecause\a—>

\a—b\foralla,6€[0,1].(Toprovethisassertionyoucannotethat

theright-handsideoftheinequalityisnon-negative,sotheinequality

isequivalentto

|a-6|>|a-6|2

(squarebothsides).Thisinequalityholdswhena=bandotherwise

isequivalentto1>\a-b|,whichwillholdwhena,b￡[0.1].

2

Economics205.Fall2010:QuizII

September3.2010

Instructions.Trytoanswerallthreeproblems.(Readallofthequestions

nowandstartontheonesthatseemeasiest.)Thinkbeforeyouwrite.You

shouldbeabletodoeverythingwithoutmuchtediouscomputation.Makeyour

answersascompleteandrigorousaspossible:givereasonsforyourcomputations

andproveyourassertions.Informalandintuitiveargumentsarebetterthan

nothing.

1.Decidewhethereachofthestatementsbelowistrue.Ifthestatementis

true,thenproveit.Ifthestatementisfalse,thengiveacounterexample.

Ineachpart/:R—>IR,istwicecontinuouslydifferentiableandstrictly

concave.

(a)x=1cannotsolveminf(x)subjecttoxe[0,1].

(b)If⑴=0,then1isalocalmaximumoff.

(c)Thereexistsnofunctionf(satisfyingtheassumptionsoftheprob-

lem)suchthat/(0)=/(I)=f⑵.

-ioo-

2.LetA=001.

_010_

(a)FindtheeigenvaluesofA.

(b)Findamaximalcardinalitysetoflinearlyindependent,eigenvectors

forA.Associatetheseeigenvectorswiththeeigenvaluesyoufound

inParta.

(c)IsAdiagonalizable?

(d)Ifthematrixisdiagonalizable,findamatrixPsuchthatA=

PDPT,whereDisdiagonal.

(e)StatewhetherthequadraticformQ(N)=xlAxispositive(semi-)

definite,negative(semi-)definite,orindefinite.

3.Letw=(1,4,0)andv=(1,0,2).

(a)Findtheequationofthelinethatpassesthroughthepointwinthe

directionv.

(b)Findtheequationofahyperplanethatcontainsthepointwand

containsthelineyoufoundinpart(a).

(c)Findanequationofalinethatiscontainedinthehyperplanethat

youfoundinpart(b),containsthepointw,andisorthogonaltothe

lineyoufoundinpart(a).

1

Economics205,Fall2010:QuizII,PossibleAnswers

September3.2010

Comments.100pointspossible,range39-99,median82,mean76.

1.Somepeopledidnotknowthedefinitionofconcavity.Apparentlysome

peopleclaimedthatf<0impliesthatfismonotonicallyincreasing.

Nope(try—x2).Concavefunctionstypicallyincreaseandthendecrease

(graphslooklikeanupside-down"U."Thethirdpartyisprobablyeasiest

ifyonjustusethedefinition.

2.Rememberthatwhenyouhaveane-valueformultiplicitykyouneedto

findklinearityindependentassociatede-vectorstodiagonalize.

3.Answerstopart(c)werenotgood,apparentlyduetotimepressure.

1.(a)False.Pickafunctionthatisstrictlyconcaveanddecreasing,for

examplef(x)=1—x2.On[0,1]thisfunctionattainsitsunique

minimumat①=1.

(b)True.Infact,itwillbeaglobalmaximum.

(c)True.Bystrictconcavity,/(I)>.5/(0)+.5/(1).

2.(a)Eigenvaluesare—1and1,themultiplicityoftheeigenvalue1istwo.

(b)Twolinearlyindependenteigenvectorsassociatedwiththeeigenvalue

1are:(1,0,0)and(0,1,1).Aneigenvectorassociatedwiththeeigen-

value—1is:(0,1,—1).

(c)Aisdiagonalizable(symmetric).

(d)OnepossiblePisthematrixwithcolumnsequaltonormalizedeigen-

'100-

vectors:P—°尖壺.Inthiscase,P-1=Pl=Pand,

100

ifD=010,thenA=PDP-1.

00-1

(e)QuadraticFormisIndefiniteSinceithaspositiveandnegativeeigen-

values.

3.Letw=(1,4,0)andv=(1,0,2).

(a)Findtheequationofthelinethatpassesthroughthepointwinthe

directionv.

Point:w=(1,4,0);Direction:v=(1,0,2).Equation:w+tv.

(b)Findtheequationofahyperplanethatcontainsthepointwand

containsthelineyoufoundinparta.

1

Point:w;Orthogonaldirection:Anythingorthogonaltov.Forex-

ample:u=(0,1,0).

Equation:"?(①一僅)=0or①2=4.Therearelots(infinitelymany)

ofalternativesolutions.

(c)Findanequationofalinethatiscontainedinthehyperplanethat

youfoundinpartb,containsthepointw,andisorthogonaltothe

lineyoufoundinparta.

Point:w;Direction:mustbeorthogonaltobothvandu.Thatis,

ifthedirectionisp=(P1,P2,P3),thenp?v=0(thisguarantees

thatthelineisorthogonaltothelineinparta)andp-u=0(this

guaranteesthatthelineisintheplanedescribedinpartb).Hence

pi+2P3=0andp?=0,soadirectionisp=(2,0,—1)andequation

forlineis:

w+tp

2

Economics205,Fall2010

QuizIII

September10,2010

Instructions.Ti*ytoanswerallpartsofbothquestions.Makeyouranswers

ascompleteandrigorousaspossible.Informalandintuitiveargumentsare

betterthannothing,butpleaseprovidecompletejustification.

1.LetK={(x^y):x2+y2<4}.

(a)ProvethatKisconvex.

(b)Showthat(3,1)隼K.

(c)FindtheequationofahyperplanethatseparatesKfrom(3,1).

(d)Showthat(%(),如)=(0,2)satisfiesx2-\-y2=4.

(e)Isitpossibletosolvetheequationx24-7/2=4foryasadifferentiable

functionofxfor(rr.y)near(0,2).Ifso.writey=Y(x)andfind

y'(o).

(f)Isitpossibletosolvetheequationx2-\-y2=4forxasadifferentiable

functionofyfor(rr,y)near(0.2).Ifso,writex=X(y)andfind

X").

2.Amonopolyfirmcaninfluencedemandbyadvertising.Ifthefirmbuys

aunitsofadvertising,itcansellqunitsatthepriceP(a,q)=a(15—q).

Thepriceofaunitsofadvertisingisaa2dollars.Itcoststhemonopolist

0q2toproducequnits.

(a)Writetheprofitfunctionofthefirm.

(b)Showthatwhena=5and3=2.5thesolutiontothemonopolist

profitmaximizationproblemistoseta=5andq=5.

(c)Isitpossibletodescribehowtheprofitmaximizingvaluesofqand

achangeasaand/3change(nearthepointinpart(b))?Ifso,

computethederivativesofqandaasfunctionsofaandBnear

=(5,).

(d)Ifaincreasesto5.01and0decreasesto2.48willthemonopolisfs

outputincrease?

1

Economics205,Fall2010

QuizIIIPossibleAnswers

September10,2009

Comments.Range：50-98,Median:73,Mean:75.Larrysaysthatmostdid

wellonthefirstquestion.Hesaidthatsomepeopleconfusedthedefinition

ofconvexityofasetwithconvexityofafunction.Thedefinitionsarerelated,

ofcourse,butdifferent.Hereportedthattherewereproblemsfiguringout

whatequationsneededtobedifferentiatedtoanswerthelastpartsofquestion

two.Istillmaintainthatitiseasierandmoreintuitivetodotheseproblems

directlyratherthanattemptingtoforcethingsintotheimplicitfunctiontheorem

formula.Thesecondquestionillustratesanimportanttechnique.

1.LetK={(x,y)：x2+y2<4}.

(a)ProvethatKisconvex.

Oneanswer:h(z)=/jsaconvexfunction(secondderivativeposi-

tive),so

(Xz+(1-A)?)2<A/+(i_A)(?)2.

Itfollowsthatif(z,y),(a/,yf)€K,and(〃,v)=(Xr+(1—X)xf,Xy+

(1-W),then

u2<Xx2+(1—A)(xz)2,

v2<AT/2-F(1-A)(y)2,

andhenceu2+v2<4.

(b)Showthat(3,1)隹K.

9+1>4

(c)FindtheequationofahyperplanethatseparatesKfrom(3,1).

Theproofoftheseparatingliyperplanetheoremusethedirection

ofthelinethatconnects(3.1)tothepointinKclosestto(3,1).

Thispointturnsouttobe(x^y)=v\4(3.1).Sothehyperplane

wouldhavenormalinthedirection(3,1)—(①,g)andpassthrough

apointonthesegmentconnecting(①,g)to(3,1).Asimplerto

describeseparatinghypcrplaneisi=2.5.EverypointinKisin

{(x^y)\x<2.5},while3>2.5.

(d)Showthat(%yo)=(0,2)satisfiesx2+y2=4.

0+4=4.

1

(e)Isitpossibletosolvetheequationx2-^-y2=4for?/asadifferentiable

functionofxfor(x,y)near(0,2).Ifso.writey=Y(x)andfind

⑵.

Itispossiblebecauseatthispointthederivativeofx2+y2with

respecttoyisnotzero.Yf(2)=0.

(f)Isitpossibletosolvetheequationx2-\-y2=4forxasadifferentiable

functionofyfor(x,y)near(0.2).Ifso,writex=andfind

X").

Itisnotpossiblebecauseatthispointthederivativeofx2+y2with

respecttoxiszero.

2.Amonopolyfirmcaninfluencedemandbyadvertising.Ifthefirmbuys

aunitsofadvertising,itcansellqunitsatthepriceF(a,q)=a(15—q).

Thepriceofaunitsofadvertisingisaa2dollars.Itcoststhemonopolist

(3q2toproducequnits.

(a)Writetheprofitfunctionofthefirm.

P(a,q)q—aa?—.

(b)Showthatwhena=5and8=.5thesolutiontothemonopolist's

profitmaximizationproblemistoseta=10andq=5.

First-orderconditions:

a(15—2q)—2/3q=0and(15—q)q—2aa=0.

Youcancheck(bydifferentiatingagain)thattheobjectivefunction

isstrictlyconcave,sofirst-orderconditionscharacterizealocalmaxi-

mum.Profitsarezeroontheboundary(qora=0),sotheequations

describeaglobalmaximum.

Theseequationsaresatisfiedatthegivenpoint(checkbysubstitu-

tion).

(c)Isitpossibletodescribehowtheprofitmaximizingvaluesofqand

achangeasaandBchange?Ifso,computethederivativesofqand

aasfunctionsofaand§near(q,a,a,0)=(5,5,5,2.5).Derivatives

withrespecttoa:

(15-2Q)OIQ-2QOI4=2Aand-2(4+0)OiQ+(15-2Q)0i4=0.

or

5￡)iQ-lODiA=10and-15D1Q+5J9M=0

soZ)iQ(5,2.5)=-.4andDM(5,2.5)=-1.2

Similarly,derivativeswithrespectto/?:

5D2Q-10V2Q=0and-15D2Q+5D2A=10.

so02Q(5,2.5)=—.8and￡>2-(5,2.5)=—.4(Soitispossible.)

2

(d)Ifaincreasesto5.01and(3decreasesto2.48willthemonopolisfs

outputincrease?

Thequestionasksfor.01(2Q—2D?Q)=.01(—.4—2(—.8))=

.01(1.2)>0,sotheanswerisyes.

3

MathematicsforEconomists

Economics205,Fall2010

GeneralInformation

Instructor:JoelSobel

Office:311Economics

OfficeHours：Afterclass

Phone:(858)534-4367

Email:jsobel@

Homepage(withlinktohandoutsforcourse):

/%7Ejsobel/205fl0/205fl0home.htm

TeachingAssistant:LawrenceSchmidt(lschmidt@)

Oi'ganization

Theclassmeetsfrom8:30to(approximately)11:00everyweekdayfromMonday,August23

throughMonday,September13,withthefollowingexception:ThereisnoclassonSeptember6.I

willalsousetimebetween11:00and11:30ifnecessaryforquizzesortostayonschedule.

Inaddition,theclassroomwillbeavailablefrom1:00-4:00forstudysessionsonmostdays.On

somedaystherewillbeorganizedproblemsessionsledbytheTA.Onotherdays,studentscanuse

theroomtoworktogetheronclassmaterial.

Description

Thiscourseisarapidoverviewoftopicsincalculus,advancedcalculus,optimization,andlinear

algebrathatarerelevanttoeconomictheory.Itprovidessomeofthenecessarymathematical

backgroundtobeginthecoregraduatesequence.Thecoursecoversalargeamountofmaterialata

relativelyhighlevelofrigor.

Ifyouhavemasteredthematerialinstandardupper-divisionanalysisandlinearalgebraclasses,

thenthisclassshouldcontainlittlethatisnew.Ifithasbeenalongtimesinceyouhaveused

calculus,thenthecoursewillbedifficult.Ifyouhaveneverusedcalculus,thenthecoursemaybe

impossible.

Toavoidmisunderstandings,letmeemphasizethattheclassisnotsimplyareviewoflower-

divisioncalculus.Nordoesitcoverall(orevenmost)ofthemathematicsusedinthecoreclasses.

Requirements

Themainevaluationwillbeathreehour,closedbook,closednotesexaminationtentatively

scheduledforThursday,September16from8:30to11:30.(Thistimebecomesofficialifthereareno

complaintstoday.)Yourgradewillbethemaximumofyourgradeonthefinalexamination,anda

weightedaverageofyourfinalexamgrade(75%),andyourperformanceonquizzes.Youmust,pass

thefinalexaminationinordertoenrollinEconomics200A.OfficiallythiscourseispartoftheFall

Quarter,soyouhavetheunusualabilitytoenrollintheclassafterityoucompletedit.

Problemsareanecessarypartoflearningthematerial.Iwillsuggestproblemsandthereare

additionalproblemspostedonthewebpage.Itisimportantthatyoufindexercisesthatareatyour

level-challenging,butnotimpossible.Ifthesuggestedproblemsaretooeasyortoohard,letme

knowandI'llfindsomethingappropriateforyou.Relevantproblemsarealsoavailableinthetexts.

Therewillbealistofproblemspostedonthewebpage.Iwillsuggestproblemsfromtextsinmost

classperiods.Youshouldattempttodothemthroughoutthecourse.

1

TextsandCourseMaterial

(SB)C.SimonandL.Blume,MathematicsforEconomists

(N)W.Novshek,MathematicsforEconomists

(D)A.Dixit,OptimizationinEconomicTheory,2ndedition

(MA)K.G.Binmore,MathematicalAnalysis

(C)K.G.Binmore,Calculus

(CH)A.Chiang,FundamentalMethodsofMathematicalEconomics

(SB)shouldbeavailableintheUniversityBookstore.Ihavecopiesofallbooks(andothers).In

additiontothesebooks,mywebpagecontainscoursenotespreparedbyJoelWatsonandme.These

areaworkinprogress,filledwitherrors,inconsistentnotation,andirrelevantmaterial.Iwillmake

anefforttoupdateandaugmentthesenotesthroughouttheclass.

Therearemanybooksthatcoverthebasicmaterialofthiscourse.Feelfreetouseanotherbook

asaprimaryreference.(Ifyouarenotsurewhetheranotherbookisadequate,thencheckwithme.)

(SB)isofficiallythetextforthecourse.Ithasthefollowingstrengths:itcontainsmanyeconomic

examples;itcoversthetopicsthatIintendtocover;itcoversothermaterialthatyoushouldknow;it

hasmanyproblemsandsolutions.Ontheotherhand,itispoorlyorganizedanditsleveloftreatment

isuneven.Mylectureswillbequitedifferentfromthetextmaterial.(N)isconcise,coversmost

ofthetopics,andhasmanyproblemsandsolutions.Itscoverageofone-variablecalculusisbrief

anditsapproachtooptimizationismechanical.(D)isaniceintroductiontooptimizationfromthe

perspectiveofeconomics.(MA)isaconciseintroductionto“advanced”one-variablecalculus.It

presentsdefinitionsandtheoremswithcareandprovidesanintroductiontoproofs.Itisslightly

moreadvancedthanthecoursewillbe.Itmaybeagoodplacetolookifthematerialinthefirst

weekseemstoeasy.(C)ismorebasicthan(MA).Ithasreasonablecoverageofmostofthetopics

ofmulti-variablecalculus.(CH)isastandardreferenceforcoursesinmathematicsforeconomists,

butIfindittoomechanical.Itmaybeagoodplacetolookifthelecturesseemdifficult.Dixit

containsmaterialrelevanttotheoptimizationtopics.

Paternalism

WhenIstartedteachingthiscourse(beforeyouwereborn),Ijustintroducedmyself,described

thetopics,andbeganteachingmatii.Gradually,Ispentmoreandmoretimetellingtheclassthings

thatIthoughtwouldhelpitadjusttothegraduateprogram.NowIhavelearnedthatthefirstday

ofclassisnotagoodtimetogetadviceand,besides,you345161hearsimilaradvicefromothers.Here

isashortlistofrecommendations.Consultthelistwhenyouareready.

1.Youcannotlearnmathematicsbyreadingabook.Itisbettertoworkproblems.Itisbetter

stilltoposeproblemsyourselfandtrytosolvethem.

2.PerformanceinEcon205isrelatedtohowmuchmathyoualreadyknow.Itisagoodpredictor

ofsuccessinfirst-yearcourses.Itisabadpredictorofthequalityofyourdissertation.

3.Thehardestpartofgraduateschoolisstartingyourresearchproject.(Inparticular,itisnot

Econ205.)

4.Nooneonthefacultywantsyoutofail.

5.BenicetoRebecca,Rafael,Nieves,

6.Youdonotneedtoknoweverythingalready.

2

7.Workandplaywithclassmates.You'lllearnmorefromthemthanyourprofessors.Some

ofthemwillbefriendsandcolleaguesforlife.

8.Figureoutwhatisimportanttoyou.

9.Goodresearchprojectsarenotscarce,buttheyarehardtofind.

TopicalOutlineandReferences

ThetableonthenextpageliststhetopicsthatIhopetocover.(Irarelyreachdifferential

equationsandintegration.)Itrelatesthetopicstopagesinfiveofthetextsmentionedabove.The

numberofpagesdevotedtoeachtopicvariesdrasticallyfromtexttotext.Thequalityandthelevel

oftreatmentvaryaswell.

TopicChMACNSB

BasicConcepts132-441-48:65-841-2;36-423-9;847-57

Continuity145-4985-912-3;42-4410-21

Differentiability128-32;149-7492-1003-522-34;39-42;70-4

MeanValueTheorems254-62101-85-6822-32

Extrema,Concavity43-6;51-69

One-variablewrapup138-4375-103

Vectors54-871-32199-204;209-30

Eigenvalues188-94;579-84;601-7;609-15

QuadraticForms375-86;398-404:620-32

VectorCalculus169-7839-5956-70273-95;301-5;313-28

Multi-variableMVT101-2970-73328-32;832-6

ImplicitFunctions184-86;204-27161-211133-46334-64

UnconstrainedOptimization231-54;307-68149-546-7;73-77375-86;396-410

EqualityConstraints369-43285-9577-103411-23:478-80

InequalityConstraints688-755131-35111-127424-78;480-2

Integration435-57226-469-19887-92

DifferentialEquations470-96313-3220633-665

3

Economics205FinalExaminationFall2010

CommentsonCourseGrade.TotalPoints:1200.High:1134;Low:558;Median:913;Mean

900.Formula:Maximumof(Final,.75Final+Quizzes,5/6Final+2BestQuizzes).Grading:

?LowestB,662.

?LowestB+,815.

?LowestA—,854

?LowestA,1023.

CommentsonFinaLHigh:1130/1200;Low492;Median895;Mean:882.

I.Somepeoplewerecasualaboutjustifyingtheirstepsandaboutthedomainofdefinition.

2.Fine(exceptafewpeopledidnotknowhowtoperformintegrationbyparts).

3.OK.

4.Somepeopledidextrawork(youneededtodiagonalizeonlyonematrix).

5.Minordeductionsfornotjustifyingyourmethod.

6.Part(b)hadatypo(correctedbelow).Youneedstrictmonotonicity(notcontinuity)for

uniqueness.Mostrespondedtothepoorlyposedquestionbywritingnonsense.Weallo-

catedallofthepointsintheproblemtotheotherparts,sonoonelostpointsforresponses

to(b).Onpart(c)severalpeopleforgotthatCEwasimplicitlydefinedbytheequation

(theytreatedIhsasCinsteadof〃(C)).Thisisasignificanterrorandledtoasignificant

deduction.

7.HereitwasokifyousolvedtheproblemusingtheobjectivefunctionSa;1^3^13—wx2—wy2

(theanswersbelowarefor—wx2—wy2.

8.Onthefirstpartsomepeopleactedasispositivesemi-definiterequiresazeroeigenvalue.

Nottrue.Positivedefinitematricesandpositivesemi-definite(inthesamewaythatpositive

numbersarenon-negative).

1.Ineachpart,determineatwhichpointsthederivativeofthefunctionhexists.Whenitdoes

exist,computeit.Whenitdoesnotexist,explainwhyitdoesnotexist.

(a)h{x)=log(l+log(l+N)).

Forcontinuityyouneedtheargumentsofthelogstobepositive.Thismeansthatyou

needx>—1inorderfor1+log(l+%)>0andlog(l+T)>—1inorderfor

1+log(l+x)>0.Thismeansyouneed1+x>e~1orx>6T—1.

h!(x)=-———

(1+x)(l+log(l+n))

bythechainrule(sincethederivativeoflog(l+a)=1/(1+7)).

1

(b)h(x)=?兩)2.

Thisoneisdifferentiablewhenrr>0(compositionofdifferentiablefunctions).Since

theformulaisjustacomplicatedwayofwritingh,⑺=x2,h![x]=2x.

(c)h(x)=J；于⑹dyforacontinuousfunction/.

Y(N)=/(/),alwaysdifferentiable(bythefundamentaltheoremofcalculus).

(d)h(x)=J：f⑺dgforacontinuousfunctionf.

Here//(①)=/⑺+//(①)providedthatfisdifferentiableatx.When力#0,

differentiabilityoffatrrisanecessaryconditionforhtobedifferentiableatx.When

z=0,"(0