2023年电大《高等数学基础》期末模拟考试试题及参考答案资料小抄

高等数学基础试题类型高等数学基础试题类型分为单项选择题、填空题、计算题和应用题。单项选择题的形式为四选一，即在每题的四个备选答案中选出一个正确答案；填空题只要求直接填写结果，不必写出计算过程和推理过程；计算题或应用题要求写出文字说明、演算步骤或推证过程。四种题型分数的百分比为：单项选择题20%，填空题20%，计算题44%，应用题16%。期末考试采用闭卷笔试形式，卷面满分为100分，考试时间为90分钟。高等数学基础模拟题一、单项选择题1.函数的图形关于（A）对称．(A)坐标原点(B)轴(C)轴(D)2.在下列指定的变化过程中，（C）是无穷小量．(A)(B)(C)(D)3.设在可导，则（C）．(A)(B)(C)(D)4.若，则（B）．(A)(B)(C)(D)5.下列积分计算正确的是（D）．(A)(B)(C)(D)6.设函数的定义域为，则函数的图形关于（A）对称．(A)(B)轴(C)轴(D)坐标原点7.当时，变量（C）是无穷小量．(A)(B)(C)(D)8.设，则（B）．(A)(B)(C)(D)9.（A）．(A)(B)(C)(D)10.下列无穷限积分收敛的是（B）．(A)(B)(C)(D)二、填空题（每小题3分，共15分）1.函数的定义域是．2.若函数，在处连续，则．3.曲线在处的切线斜率是3．4.函数的单调增加区间是．5.若，则．6.函数的定义域是．7.函数的间断点是．8.曲线在处的切线斜率是．9.函数的单调减少区间是．10.．三、计算题（每小题11分，共44分）1.计算极限．解：2.设，求．解：3.计算不定积分．解：由换元积分法得4.计算定积分．解：由分部积分法得5.计算极限．解：．6.设，求．解：7.计算不定积分．解：8.计算定积分．解：四、应用题（本题16分）1、某制罐厂要生产一种体积为V的有盖圆柱形容器，问容器的底半径与高各为多少时用料最省？解：设容器的底半径为，高为，则其表面积为由，得唯一驻点，由实际问题可知，当时可使用料最省，此时，即当容器的底半径与高分别为与时，用料最省．2、圆柱体上底的中心到下底的边沿的距离为l，问当底半径与高分别为多少时，圆柱体的体积最大？解：如图所示，圆柱体高h，与底半径r满足圆柱体的体积公式为将代入得求导得令得，并由此解得，即当底半径为，高时，圆柱体得体积最大。典型例题例1计算极限．解利用重要极限，及极限的运算法则得例2计算极限．解：利用极限的运算法则得例3设，求．解：利用导数的运算法则得例4设，求．解：设，得利用复合函数求导法则，得例5设是由方程确定的函数，求．解：利用导数运算法则和复合函数求导法则，等式两端分别对求导得左：右：由此得整理得由微分定义得例6计算．解：利用换元积分法得例7计算．解：利用分部积分法得例8求曲线上的点，使其到点的距离最短．解：曲线上的点到点的距离公式为与在同一点取到最小值，为计算方便求的最小值点，将代入得令令得．可以验证是的最小值点，并由此解出，即曲线上的点和点到点的距离最短．高等数学基础第一次作业单项选择题⒈下列各函数对中，（C）中的两个函数相等．A.，B.，C.，D.，⒉设函数的定义域为，则函数的图形关于（C）对称．A.坐标原点B.轴C.轴D.⒊下列函数中为奇函数是（B）．A.B.C.D.⒋下列函数中为基本初等函数是（C）．A.B.C.D.⒌下列极限存计算不正确的是（D）．A.B.C.D.⒍当时，变量（C）是无穷小量．A.B.C.D.⒎若函数在点满足（A），则在点连续。A.B.在点的某个邻域内有定义C.D.（二）填空题⒈函数的定义域是．⒉已知函数，则x2-x．⒊．⒋若函数，在处连续，则e．⒌函数的间断点是．计算题⒈设函数求：．解：，，⒉求函数的定义域．解：有意义，要求解得，则定义域为⒊在半径为的半圆内内接一梯形，梯形的一个底边与半圆的直径重合，另一底边的两个端点在半圆上，试将梯形的面积表示成其高的函数．解：AROhEBC设梯形ABCD即为题中要求的梯形，设高为h，即OE=h，下底CD＝2R直角三角形AOE中，利用勾股定理得则上底＝故⒋求．解：＝⒌求．解：⒍求．解：⒎求．解：⒏求．解：⒐求．解：⒑设函数讨论的连续性，并写出其连续区间．解：分别对分段点处讨论连续性（1）所以，即在处不连续（2）所以即在处连续由（1）（2）得在除点外均连续故的连续区间为《高等数学基础》第四次作业（一）单项选择题⒈若的一个原函数是，则（D）．A.B.C.D.⒉下列等式成立的是（D）．AB.C.D.⒊若，则（B）．A.B.C.D.⒋（B）．A.B.C.D.⒌若，则（B）．A.B.C.D.⒍由区间上的两条光滑曲线和以及两条直线和所围成的平面区域的面积是（C）．A.B.C.D.（二）填空题⒈函数的不定积分是．⒉若函数与是同一函数的原函数，则与之间有关系式．⒊⒋⒌若，则⒍3⒎若无穷积分收敛，则（三）计算题⒈⒉⒊⒋⒌⒍⒎⒏（四）证明题⒈证明：若在上可积并为奇函数，则．证:证毕⒉证明：若在上可积并为偶函数，则．证：⒊证明：证：=《高等数学基础》第二次作业（一）单项选择题⒈设且极限存在，则（C）．A.B.C.D.cvx⒉设在可导，则（D）．A.B.C.D.⒊设，则（A）．A.B.C.D.⒋设，则（D）．A.B.C.D.⒌下列结论中正确的是（C）．A.若在点有极限，则在点可导．B.若在点连续，则在点可导．C.若在点可导，则在点有极限．D.若在点有极限，则在点连续．（二）填空题⒈设函数，则0．⒉设，则．⒊曲线在处的切线斜率是⒋曲线在处的切线方程是⒌设，则⒍设，则（三）计算题⒈求下列函数的导数：⑴⑵⑶⑷⑸⑹⑺⑻⒉求下列函数的导数：⑴⑵⑶⑷⑸⑹⑺⑻⑼⑽⑾⒊在下列方程中，是由方程确定的函数，求：⑴⑵⑶⑷⑸⑹⑺⑻⒋求下列函数的微分：⑴⑵⑶⑷两边对数得：⑸⑹⒌求下列函数的二阶导数：⑴⑵⑶⑷（四）证明题设是可导的奇函数，试证是偶函数．证：因为f(x)是奇函数所以两边导数得：所以是偶函数。《高等数学基础》第三次作业（一）单项选择题⒈若函数满足条件（D），则存在，使得．A.在内连续B.在内可导C.在内连续且可导D.在内连续，在内可导⒉函数的单调增加区间是（D）．A.B.C.D.⒊函数在区间内满足（A）．A.先单调下降再单调上升B.单调下降C.先单调上升再单调下降D.单调上升⒋函数满足的点，一定是的（C）．A.间断点B.极值点C.驻点D.拐点⒌设在内有连续的二阶导数，，若满足（C），则在取到极小值．A.B.C.D.⒍设在内有连续的二阶导数，且，则在此区间内是（A）．A.单调减少且是凸的B.单调减少且是凹的C.单调增加且是凸的D.单调增加且是凹的（二）填空题⒈设在内可导，，且当时，当时，则是的极小值点．⒉若函数在点可导，且是的极值点，则0．⒊函数的单调减少区间是．⒋函数的单调增加区间是⒌若函数在内恒有，则在上的最大值是．⒍函数的拐点是x=0．（三）计算题⒈求函数的单调区间和极值．令X2(2,5)5+极大-极小+y上升27下降0上升列表：极大值：极小值：⒉求函数在区间内的极值点，并求最大值和最小值．令：⒊试确定函数中的，使函数图形过点和点，且是驻点，是拐点．解：⒋求曲线上的点，使其到点的距离最短．解：，d为p到A点的距离，则：⒌圆柱体上底的中心到下底的边沿的距离为，问当底半径与高分别为多少时，圆柱体的体积最大？设园柱体半径为R，高为h，则体积⒍一体积为V的圆柱体，问底半径与高各为多少时表面积最小？设园柱体半径为R，高为h，则体积答：当时表面积最大。⒎欲做一个底为正方形，容积为62.5立方米的长方体开口容器，怎样做法用料最省？解：设底连长为x，高为h。则：侧面积为：令答：当底连长为5米，高为2.5米时用料最省。（四）证明题⒈当时，证明不等式．证：由中值定理得：⒉当时，证明不等式．请您务必删除一下内容，O(∩_∩)O万分谢谢！！！2023年中央电大期末复习考试小抄大全，电大期末考试必备小抄，电大考试必过小抄IndonesiahasemergedasatopvacationdestinationforChinesethissummer,alongwithThailand'sPhuketislandandtheMaldivesarchipelago.EncouragedbygrowthinChinesetravelerstoIndonesia,thegovernmentinJakartahasrecentlyrelaxeditsvisapolicy.SinceJune10,ChinesetouristscanenterIndonesiathroughnineappointedlocations,includingtheSoekarnoHattaInternationalAirportinJakarta,theNgurahRaiInternationalAirportinBaliandtheKualaNamuInternationalAirportinMedan,byjustgettingtheirpassportsstampedonarrival.Officially,itiscalledthe"free-visa"scheme.ThenewpolicyalsomeansChinesetouristscansave$35onvisafees,whichwilllikelymaketripstoIndonesiamorealluring.WhilesuchanarrangementallowsChinesevisitorstostayupto30daysforthepurposeoftraveling,thoseseekingextensionswillneedtogetpaidvisasfromthatcountry'smissionsinBeijing,

Shanghai,

Guangzhou

or

HongKong,accordingtoanofficialattheIndonesiaembassyinBeijing.SantoDarmosumarto,headoftheembassy'sinformation,socialandculturalsection,saysthenewpolicyisaimedatstrengtheningpeople-to-peoplecontactsbetweenthetwocountries.DaiYu,marketingdirectorofCtrip,amajorChineseonlinetravelagency,says:"We'veseenanearly50percentgrowthoverthepreviousmonthinthenumberofChinesetouriststoBalisincethepolicywasannounced."DuringhistriptoChinainMarch,IndonesianPresidentJokoWidodoproposedthathiscountryandChinaaimtoincreasetwo-wayvisitstoamaximumof10millionpeopleinthenextfewyears.TheIndonesiangovernmenthassettheirsightsongreeting2millionChinesetouristsbytheendofthisyear,accordingtotheofficial."TheIndonesiangovernmenthopesthatIndonesiannationalswouldalsobegivenpreferentialtreatmentwhilevisitingChina,"saysDarmosumarto.In2023,thenumberofChinesetouristswhovisitedIndonesiastoodaround807,000.Lastyear,itincreasedto959,000,headds.BaliremainsthemostpopularsiteamongChinesetourists.InFebruary,morethan92,200ChinesetouristsmadetripstoBali,rankingfirstamongoverseasvisitors,theBaliTimesreported.Lastyear,theislandwitnessedmorethan586,000arrivalsfromChina,anincreaseof51percentover2023."Baliisn'tjustforsightseeing,butalsoforweddings,honeymoons,golfandmanyoutdooractivities,"saysFanWenqing,amarketingexecutiveatGaruda,Indonesia'snationalairliner.GarudaoffersthreenonstopflightsbetweenBeijingandBali,andfoureachweekbetweenBeijingandJakarta.TherearedailyflightsfromGuangzhou,in

Guangdong

province,andShanghaitoJakartaaswell.IttakesfromsixtoeighthoursonanonstopflighttoreachIndonesiafromChina.Indonesiaishometomorethan17,000islands.TouristscantourYogyakartatoseePrambanamandBorobudur,twohistoricalandreligioussites,andexplorethelocalhandicraftmarkets.Thebaseofavolcano,Bandung,whichisalsoknownasthe"ParisofJava",isidealforadventure-seekers.Manyecotourismdestinations,suchasTobaLakeinSumatra,TanjungPutingNationalParkinKalimantanandBunakeninManado,areplacestheembassyofficialrecommendsfortourists.Indonesiaboastsworld-classsurfingfacilitiesalongtheshoresofSumatradowntoNusaTenggaraIslands.Recently,Lombok,asisterislandeastofBalihasgottensomuchattentionfromChinesetravelersthatsometravelagenciesareofferingtwinpackagesforBaliandLombok.ThebesttimetovisitBaliisfromMaytoOctober,whenitdoesn't

rain

muchandtheweatheriscool.Atpresstime,aweeklongpackagefromShanghaiorGuangzhoutoBaliwasalittlemorethan3,000yuan($480)onCtrip'swebsite.ChinesetouristshavetakenashinetoraftinginUbud,sightseeinginTanahLot,andsurfinganddivinginNusaLembongan,saysDai.Civetcoffee,essentialoilandwoodcarvingsareamongthemostpopularproductsforChinesetouristsinIndonesia,shesays.Indonesianfoodisalsosomethingthatvisitorsshouldn'tmiss.Localcuisinesfeaturemanyspices,includingpeppersandcloves,andcoconutpulp.Friedrice,"dirtyduck"androastsucklingpigarerecommended.Children'sillustrationbooksfirstappearedinthe1600sinEurope.TheirpassagetoChina,however,tookmuchlonger,roughly300years.InChina,children'sillustrationbookshaveahistoryofjust100yearsandtheirdevelopmentcanbedividedintofourstages.Fromearly1900stomid1900s,alargenumberofbooksandmagazineswithillustrationsappearedwiththefastdevelopmentofthemodernpublishingindustry.Theillustration

artsatthattimewereamixedgenreofChinesetraditional

paintingsandthewestern

style,andmostofthestoriesoriginatefromthefundamentalchangesthatChinaexperiencedafterthecollapseofQingDynasty(1644-1911),thewarlordstrugglesandthewaragainsttheJapaneseinvasion.Thesecondstagewasfrommid1900stolate1970s,whenthe"culturalrevolution"(1966-76)ended.Thefastdevelopmentofsomespecialpublishinghousesforchildrengavebirthtoalargenumberofillustration-storybooks,whichmadeagooduseoffolkarttechniques,suchaswoodboardcarving,frescoes,andinkandwashpainting,etc.ManyofthepopularstoriesarefromChinesehistoryandwarsafter1900s.Thethirdstageisfromearly1980stolate1990s,agoldenperiodforthedevelopmentofchildren'sillustrationbooks.Thereformandopening-upbroughtinnewconceptsandideasfromtheWesttoChina,acountrythathadbeenclosedtotheWestfornearlythreedecades.Fastdevelopmentofeconomy,societyandcultureprovidedthewritersandpainterswithalotofinspirationtocreatenewliteratureworksanddrawillustrations.Chinaalsoimportedlargeamountsofchildren'sillustrationbooksfromtheWestandJapan.ManyyoungparentsinChinahaveastrongnostalgiaaboutchildren'sillustrationbookstheyreadinthe1980sand1990s.ThisisalsothelasterainChina'sartisticinnovationbeforetheadventoftheInternetperiod.ThefourthstageisfeaturedwiththespreadoftheInternetandtheothercomputerizedmeansofartinnovation.Thehand-drawnchildren'sillustrationbooksaregraduallyreplacedbycartoonsdrawnandcopiedbysoftwareandmachines,andshownontablets,computers,televisionandsmartphones,insteadofbooks.HainanTropicalWildlifeParkandBotanicalGardenisnature'shaven,withnolessthan4,000rarebirdsandanimalsrepresenting200species.Withtropicalforestcoveringover90percentofitsarea,thegardenoffersbeautifulviewsofawiderangeoftropicalplants,flowersandfruittrees.Thescenicspotabout25kilometersfromthecenterofHaikouoffersadrive-thrusafaritourwhereyoucangetupclosewithelephants,lionsandbears.The"SafarionFoot"areafeatureswalkingtrailswherevisitorscanseetherestoftheanimals,includinghipposandmonkeys,andfeedthemforasmallfee.Visitorsalsocanseewhathappenswhenyoubreedalionandatiger.Twouniqueattractionsarerarehybrids:aliger,theoffspringofamallionandafemaletigerandatigon,whichiscreatedbyafemalelionandamaletiger.Asa

teachingcenteroftheNationalPopularScienceEducationprogram,theparkalsoservesasasecondclassroomforstudentstolearnaboutwildlifeandplants.Sinceitwasestablishedin1995,ithasreceivedmorethanonemillionstudentvisitors.AtacrucialmomentwhenpessimismhasbroughttheGreekcrisistosuchaheadthatthecountryisclosertoadefaultonitsdebtrepaymentsthanever,LiofferedasparkofenthusiasmtorestartthestalledtalksbycallingonGreeceanditscreditorstoreachalast-minutedealthatwillallowGreecetoremainintheeurozone.Giventheescalationofthecountry'scrisisovertheweekendsentashockwavethroughtheglobalstockmarketsonMonday,thereareamplereasonstoworryabouttheuncertaintiessurroundingtheeventsinGreeceandtheeffectadefaultwouldhaveonEurope,relationsbetweenChinaandtheEU,globalfinancialstabilityandthenascentglobaleconomicrecovery.Greecehasannouncedthatitwillholdareferendumonabailoutplanproposedlastweekbythecountry'screditors.Inresponse,Greece'seurozonepartnershaverefusedtoextendthecountry'sbailoutprogramandtheEuropeanCentralBankcappeditsemergencysupportforthecountry'sbanks.ArealdangernowloomsofGreeceexitingtheeuro,whichmightderailtheglobaleconomicrecoveryanddamagethelong-termviabilityoftheeuroasacurrency.However,suchatragedyisneithercertainnorunavoidable.TheEUshouldshoulderitsresponsibilitytopreventtheGreekcrisisfromovershadowingthefragileglobalrecovery.Asamajorcustomerandsupplierofthe28-nationEU,andaresponsiblelong-termholderofEurobonds,China'sconfidenceinandcommitmenttoastrongeurozoneoffersEUleadersthenecessarysupporttolookattheGreekcrisisfromabroaderandlongerperspective.WhenLisaidthatChinawillnotonlyco