2015年電大《高等數(shù)學(xué)基礎(chǔ)》期末模擬考試試題及參考答案資料小抄

高等數(shù)學(xué)基礎(chǔ)試題類型高等數(shù)學(xué)基礎(chǔ)試題類型分為單項選擇題、填空題、計算題和應(yīng)用題。單項選擇題的形式為四選一，即在每題的四個備選答案中選出一個正確答案；填空題只要求直接填寫結(jié)果，不必寫出計算過程和推理過程；計算題或應(yīng)用題要求寫出文字說明、演算步驟或推證過程。四種題型分數(shù)的百分比為單項選擇題20，填空題20，計算題44，應(yīng)用題16。期末考試采用閉卷筆試形式，卷面滿分為100分，考試時間為90分鐘。高等數(shù)學(xué)基礎(chǔ)模擬題一、單項選擇題1函數(shù)的圖形關(guān)于（A）對稱2EXYA坐標原點B軸C軸DYXY2在下列指定的變化過程中，（C）是無窮小量ABCD1SINX01SINX01LNE1X3設(shè)在可導(dǎo)，則（C）F0HFFH2LM0ABCDX2X0X20XF4若，則（B）CFFDFDLN1ABCDLNXXLCXFLCXF15下列積分計算正確的是（D）ABCD0DSI11DE0XD2SIN00DOS16設(shè)函數(shù)的定義域為，則函數(shù)的圖形關(guān)于（A）對稱XF,XFAB軸C軸D坐標原點YY7當時，變量（C）是無窮小量ABCDX1XSIN1EX28設(shè)，則（B）FEFFLIM0ABCD2E4E29（A）XFDABCD2FXFD121XFXFD210下列無窮限積分收斂的是（B）ABCD0DEX0EX1DX1二、填空題（每小題3分，共15分）1函數(shù)的定義域是241LNXY2,2若函數(shù)，在處連續(xù)，則012XKXFKE3曲線在處的切線斜率是33F,4函數(shù)的單調(diào)增加區(qū)間是YARCTN,5若，則XFSIDXFXSIN6函數(shù)的定義域是1L92|13,27函數(shù)的間斷點是0SINXY0X8曲線在處的切線斜率是1F2,29函數(shù)的單調(diào)減少區(qū)間是2Y,110XDSINSINC三、計算題（每小題11分，共44分）1計算極限解1ILM2X21SINLM1SINL12XXX2設(shè)，求Y3ECOSYD解3DECOSDXXXL3DESIXXLNIN3計算不定積分XE21解由換元積分法得CUXED1ED21X14計算定積分E1LNX解由分部積分法得E1LNL1EX5計算極限解XX5SIN6LM000SI6SI6LLM55XX6設(shè)，求Y2Y解ISIN2LXXSINCO2LNX7計算不定積分D3CO解SISI3ISDXXXXN1ID3N1COS3C8計算定積分E1DL2X解EE112LND2LNDLXX221LN1|E5四、應(yīng)用題（本題16分）1、某制罐廠要生產(chǎn)一種體積為V的有蓋圓柱形容器，問容器的底半徑與高各為多少時用料最省解設(shè)容器的底半徑為，高為，則其表面積為RHHRS224V由，得唯一駐點，由實際問題可知，當時可使用料最省，此時，即當容器的底0S32R32VR34VH半徑與高分別為與時，用料最省3342、圓柱體上底的中心到下底的邊沿的距離為L，問當?shù)装霃脚c高分別為多少時，圓柱體的體積最大解如圖所示，圓柱體高H，與底半徑R滿足22HRL圓柱體的體積公式為V將代入得22RL2LH求導(dǎo)得23HL令得，并由此解得，即當?shù)装霃綖?，高時，圓柱體得體積最大。0V3L6R63RL3HL典型例題例1計算極限321SINLM1XX解利用重要極限，及極限的運算法則得I013SINL2SNLI11XXXX1SIN3LIM1XXLILIM1X4例2計算極限27623解利用極限的運算法則得54LI2LI176LIM3323XXXXX例3設(shè)，求YSINY解利用導(dǎo)數(shù)的運算法則得XXXXY2333SINSINLLSINLX233SICOLLXX232SICOSLN1例4設(shè)，求YY解設(shè)，得2INUVLUSIN2XV利用復(fù)合函數(shù)求導(dǎo)法則，得XVXYYSIL2XVUSILUCO122TASINCO例5設(shè)是由方程確定的函數(shù)，求4ELYXDY解利用導(dǎo)數(shù)運算法則和復(fù)合函數(shù)求導(dǎo)法則，等式兩端分別對求導(dǎo)得X左YX1LLN右YXXEEE444YX34E由此得整理得YYX3141X由微分定義得D4E例6計算X21解利用換元積分法得1DE1DE221XXXCUCX1E例7計算DLN解利用分部積分法得LND1LN1LL1XXXXN11XXDLCX1LN1例8求曲線上的點，使其到點的距離最短Y20,2A解曲線上的點到點的距離公式為XD與在同一點取到最小值，為計算方便求的最小值點，將代入得22DXY2令22XD令得可以驗證是的最小值點，并由此解出，即曲線上的點和點011X2D2YXY22,1到點的距離最短,1,A高等數(shù)學(xué)基礎(chǔ)第一次作業(yè)（一）單項選擇題下列各函數(shù)對中，（C）中的兩個函數(shù)相等A2XF，XGB2XF，XGC3LNF，LND1F，12設(shè)函數(shù)的定義域為,，則函數(shù)F的圖形關(guān)于（C）對稱A坐標原點BX軸CY軸DXY下列函數(shù)中為奇函數(shù)是（B）A1LN2YBXCOSC2XAD1LNXY下列函數(shù)中為基本初等函數(shù)是（C）AXBYC2YD0,1XY下列極限存計算不正確的是（D）A12LIMXB01LNIM0XXCSINLMXD01SINLMXX當0時，變量（C）是無窮小量ASINBCSID2LN若函數(shù)XF在點0滿足（A），則XF在點0連續(xù)。ALI0FB在點的某個鄰域內(nèi)有定義CM0FXDLIMLI00XFFXX（二）填空題函數(shù)1LN392XF的定義域是|3已知函數(shù)，則XFX2XXX21LIM若函數(shù)0,1XKF，在處連續(xù)，則KE函數(shù),SINXY的間斷點是X（二）計算題設(shè)函數(shù)0,EXXF求1,2F解2F，0F，1FE求函數(shù)1LGXY的定義域解21LGXY有意義，要求20X解得102X或，則定義域為1|02X或在半徑為R的半圓內(nèi)內(nèi)接一梯形，梯形的一個底邊與半圓的直徑重合，另一底邊的兩個端點在半圓上，試將梯形的面積表示成其高的函數(shù)解DAROHEBC設(shè)梯形ABCD即為題中要求的梯形，設(shè)高為H，即OEH，下底CD2R直角三角形AOE中，利用勾股定理得22AEORH則上底2AERH故22HSRHRA求XXSIN3LM0解000SISIN3LILLM22XXX132求1SINL21X解11LMLILISNISNXXX求X3TANLI0解00SI1SIN31LLM3COCOSXXXA求XSIN1LM20解222200011LLILIMISIN1SINXXXXX02LIIX求XX31LIM解1433111LIMLILIMLI333XXXXXE求4586LI24XX解422MLILIM113XXX設(shè)函數(shù)1,1,2XXF討論F的連續(xù)性，并寫出其連續(xù)區(qū)間解分別對分段點,處討論連續(xù)性（1）11LIMLI10XXF所以1LILIXXFF，即FX在1處不連續(xù)（2）2211LILIMXXFF所以11LILIXXFFF即FX在1處連續(xù)由（1）（2）得在除點1外均連續(xù)故F的連續(xù)區(qū)間為,高等數(shù)學(xué)基礎(chǔ)第四次作業(yè)（一）單項選擇題若XF的一個原函數(shù)是X1，則F（D）ALNB2CD32X下列等式成立的是（D）ADXFFBDXFFCDFFDDXFFX若COS，則（B）AINBCCCSINDCOSXFXD32（B）A3FB32FC1XFD31XF若CXFFD，則XFD1（B）AB2CCF2DCXF1由區(qū)間,BA上的兩條光滑曲線XFY和G以及兩條直線A和B所圍成的平面區(qū)域的面積是（C）AAXGFDBBAFGDCBAXGFDDAXGFD（二）填空題函數(shù)F的不定積分是XF若函數(shù)XF與G是同一函數(shù)的原函數(shù)，則XF與G之間有關(guān)系式。CXGF常數(shù)DE22TANCT若XXF3OS，則XF3COS9X35D21SI3若無窮積分1XP收斂，則0P（三）計算題CXDXX1SINCOSDCOS2EE2ECXDXXLNLN1LCXXD2SIN41O2OS2CS2SIE11E1LN3LN3LDL3EXX414222021002DEXXLNDL12E1XEEEXDX121E2L（四）證明題證明若XF在,A上可積并為奇函數(shù)，則0DAXF證AAATFTDTFDXFT令0AAAXDF證畢證明若X在,上可積并為偶函數(shù)，則AAXFXF0D2D證AAAXFXFXF00DDAAXFTFTT0,是偶函數(shù)則令證畢AAAAXFFFF0000D2D證明XXXDD證AAAAAFXFFFF0000XXX00證畢高等數(shù)學(xué)基礎(chǔ)第二次作業(yè)（一）單項選擇題設(shè)0F且極限XFLIM0存在，則XFLI0（C）ABFCDCVX設(shè)XF在0可導(dǎo)，則HFFH2LI00（D）A2BXCXD0XF設(shè)XFE，則FF1LIM0（A）AB2CEDE4設(shè)91XXF，則0F（D）A9BCD9下列結(jié)論中正確的是（C）A若XF在點0有極限，則在點0可導(dǎo)B若XF在點0連續(xù)，則在點0X可導(dǎo)C若在點可導(dǎo)，則在點X有極限D(zhuǎn)若在點有極限，則在點連續(xù)（二）填空題設(shè)函數(shù)0,01SIN2XXF，則F0設(shè)XXFE5E2，則FDLX5LN2曲線1在,處的切線斜率是1K曲線XFSIN在4處的切線方程是42XY設(shè)Y2，則YLN2XX設(shè)L，則1（三）計算題求下列函數(shù)的導(dǎo)數(shù)YXYE3XXEY2123LNCOT2LNCSXLXY2L3SY42COS32LSINXXXSINL2Y2SINL1IYL4XXLCO43X3SI2XXY223LNSICSYLNTAEEEX1OSTAN求下列函數(shù)的導(dǎo)數(shù)Y21X212XX3COSLNY323TANCOSINXYX87X81Y3Y231321XYXECOS2SINXE22XYNYISINCOSSI1XN2SI5X2IN5LXXY2SINEXEY2SINSI22XX22LXXEXXYEEXXXEEYXN在下列方程中，是由方程確定的函數(shù)，求Y2COSEXYY2INYEX2COSINL1COSSILNIYX2SIN2ICO2YXYXYXYSIN2COS2222COSINXYXL1Y12ELNXYYEXYXSI12XXYEYINCOYXCOS2IN3XY223YEXY5LNLYXY2LN15YX求下列函數(shù)的微分DCSOTDXXSICO22XYSINLDY2INLS11ARCDXXDXXDY22221131兩邊對數(shù)得1LNL3LNXY13XY13XYESI2DEDXEDX2SISI23TANXX2CC33求下列函數(shù)的二階導(dǎo)數(shù)YLXN1XY1SIYCOXCOS2SINXYARCTN2121X3X3LN2Y223LN23LN4XXY（四）證明題設(shè)XF是可導(dǎo)的奇函數(shù)，試證F是偶函數(shù)證因為FX是奇函數(shù)所以X兩邊導(dǎo)數(shù)得1FFFXF所以F是偶函數(shù)。高等數(shù)學(xué)基礎(chǔ)第三次作業(yè)（一）單項選擇題若函數(shù)XF滿足條件（D），則存在,BA，使得ABFFA在,BA內(nèi)連續(xù)B在,內(nèi)可導(dǎo)C在,內(nèi)連續(xù)且可導(dǎo)D在,BA內(nèi)連續(xù)，在,內(nèi)可導(dǎo)函數(shù)142XF的單調(diào)增加區(qū)間是（D）A,B,C,2D,2函數(shù)52Y在區(qū)間6內(nèi)滿足（A）A先單調(diào)下降再單調(diào)上升B單調(diào)下降C先單調(diào)上升再單調(diào)下降D單調(diào)上升函數(shù)XF滿足0F的點，一定是XF的（C）A間斷點B極值點C駐點D拐點設(shè)F在,BA內(nèi)有連續(xù)的二階導(dǎo)數(shù)，,0BA，若XF滿足（C），則XF在0取到極小值A(chǔ)00XFXB0XFC,FD,0F設(shè)在內(nèi)有連續(xù)的二階導(dǎo)數(shù)，且X，則XF在此區(qū)間內(nèi)是（A）A單調(diào)減少且是凸的B單調(diào)減少且是凹的C單調(diào)增加且是凸的D單調(diào)增加且是凹的（二）填空題設(shè)XF在,BA內(nèi)可導(dǎo)，,0BAX，且當0X時F，當0時F，則0X是F的極小值點若函數(shù)F在點0可導(dǎo)，且0是F的極值點，則0XF0函數(shù)1LN2XY的單調(diào)減少區(qū)間是,函數(shù)EF的單調(diào)增加區(qū)間是若函數(shù)在,BA內(nèi)恒有XF，則XF在,BA上的最大值是AF函數(shù)352XF的拐點是X0（三）計算題求函數(shù)21Y的單調(diào)區(qū)間和極值令XX5,2駐點列表極大值27F極小值05求函數(shù)3YX在區(qū)間,0內(nèi)的極值點，并求最大值和最小值令。XXY駐點1026F最大值21最小值試確定函數(shù)DCXBAY3中的DCBA,，使函數(shù)圖形過點4,2和點10,，且2X是駐點，1X是拐點解BAC2604824163DCBA求曲線XY上的點，使其到點0,2A的距離最短解上的點是設(shè)P,，D為P到A點的距離，則XD22101XX令。AXY的距離最短到點上點0,12圓柱體上底的中心到下底的邊沿的距離為L，問當?shù)装霃脚c高分別為多少時，圓柱體的體積最大設(shè)園柱體半徑為R，高為H，則體積LHV22LHLH。33032令。L時其體積最大當,33一體積為V的圓柱體，問底半徑與高各為多少時表面積最小設(shè)園柱體半徑為R，高為H，則體積222RVSH表面積33204。令34VH答當2R3VH時表面積最大。欲做一個底為正方形，容積為625立方米的長方體開口容器，怎樣做法用料最省解設(shè)底連長為X，高為H。則225656XHXX2,22,55,Y極大極小Y上升27下降0上升2130FF側(cè)面積為XHXS25042令510532答當?shù)走B長為5米，高為25米時用料最省。（四）證明題當X時，證明不等式LNX證由中值定理得011LLX。XX時當0L1LN當0時，證明不等式EXEF設(shè)00FXFX單調(diào)上升且時當時當證畢即1,F請您務(wù)必刪除一下內(nèi)容，O_O萬分謝謝2015年中央電大期末復(fù)習(xí)考試小抄大全，電大期末考試必備小抄，電大考試必過小抄INDONESIAHASEMERGEDASATOPVACATIONDESTINATIONFORCHINESETHISSUMMER,ALONGWITHTHAILANDSPHUKETISLANDANDTHEMALDIVESARCHIPELAGOENCOURAGEDBYGROWTHINCHINESETRAVELERSTOINDONESIA,THEGOVERNMENTINJAKARTAHASRECENTLYRELAXEDITSVISAPOLICYSINCEJUNE10,CHINESETOURISTSCANENTERINDONESIATHROUGHNINEAPPOINTEDLOCATIONS,INCLUDINGTHESOEKARNOHATTAINTERNATIONALAIRPORTINJAKARTA,THENGURAHRAIINTERNATIONALAIRPORTINBALIANDTHEKUALANAMUINTERNATIONALAIRPORTINMEDAN,BYJUSTGETTINGTHEIRPASSPORTSSTAMPEDONARRIVALOFFICIALLY,ITISCALLEDTHE“FREEVISA“SCHEMETHENEWPOLICYALSOMEANSCHINESETOURISTSCANSAVE35ONVISAFEES,WHICHWILLLIKELYMAKETRIPSTOINDONESIAMOREALLURINGWHILESUCHANARRANGEMENTALLOWSCHINESEVISITORSTOSTAYUPTO30DAYSFORTHEPURPOSEOFTRAVELING,THOSESEEKINGEXTENSIONSWILLNEEDTOGETPAIDVISASFROMTHATCOUNTRYSMISSIONSINBEIJING,SHANGHAI,GUANGZHOUORHONGKONG,ACCORDINGTOANOFFICIALATTHEINDONESIAEMBASSYINBEIJINGSANTODARMOSUMARTO,HEADOFTHEEMBASSYSINFORMATION,SOCIALANDCULTURALSECTION,SAYSTHENEWPOLICYISAIMEDATSTRENGTHENINGPEOPLETOPEOPLECONTACTSBETWEENTHETWOCOUNTRIESDAIYU,MARKETINGDIRECTOROFCTRIP,AMAJORCHINESEONLINETRAVELAGENCY,SAYS“WEVESEENANEARLY50PERCENTGROWTHOVERTHEPREVIOUSMONTHINTHENUMBEROFCHINESETOURISTSTOBALISINCETHEPOLICYWASANNOUNCED“DURINGHISTRIPTOCHINAINMARCH,INDONESIANPRESIDENTJOKOWIDODOPROPOSEDTHATHISCOUNTRYANDCHINAAIMTOINCREASETWOWAYVISITSTOAMAXIMUMOF10MILLIONPEOPLEINTHENEXTFEWYEARSTHEINDONESIANGOVERNMENTHASSETTHEIRSIGHTSONGREETING2MILLIONCHINESETOURISTSBYTHEENDOFTHISYEAR,ACCORDINGTOTHEOFFICIAL“THEINDONESIANGOVERNMENTHOPESTHATINDONESIANNATIONALSWOULDALSOBEGIVENPREFERENTIALTREATMENTWHILEVISITINGCHINA,“SAYSDARMOSUMARTOIN2013,THENUMBEROFCHINESETOURISTSWHOVISITEDINDONESIASTOODAROUND807,000LASTYEAR,ITINCREASEDTO959,000,HEADDSBALIREMAINSTHEMOSTPOPULARSITEAMONGCHINESETOURISTSINFEBRUARY,MORETHAN92,200CHINESETOURISTSMADETRIPSTOBALI,RANKINGFIRSTAMONGOVERSEASVISITORS,THEBALITIMESREPORTEDLASTYEAR,THEISLANDWITNESSEDMORETHAN586,000ARRIVALSFROMCHINA,ANINCREASEOF51PERCENTOVER2013“BALIISNTJUSTFORSIGHTSEEING,BUTALSOFORWEDDINGS,HONEYMOONS,GOLFANDMANYOUTDOORACTIVITIES,“SAYSFANWENQING,AMARKETINGEXECUTIVEATGARUDA,INDONESIASNATIONALAIRLINERGARUDAOFFERSTHREENONSTOPFLIGHTSBETWEENBEIJINGANDBALI,ANDFOUREACHWEEKBETWEENBEIJINGANDJAKARTATHEREAREDAILYFLIGHTSFROMGUANGZHOU,INGUANGDONGPROVINCE,ANDSHANGHAITOJAKARTAASWELLITTAKESFROMSIXTOEIGHTHOURSONANONSTOPFLIGHTTOREACHINDONESIAFROMCHINAINDONESIAISHOMETOMORETHAN17,000ISLANDSTOURISTSCANTOURYOGYAKARTATOSEEPRAMBANAMANDBOROBUDUR,TWOHISTORICALANDRELIGIOUSSITES,ANDEXPLORETHELOCALHANDICRAFTMARKETSTHEBASEOFAVOLCANO,BANDUNG,WHICHISALSOKNOWNASTHE“PARISOFJAVA“,ISIDEALFORADVENTURESEEKERSMANYECOTOURISMDESTINATIONS,SUCHASTOBALAKEINSUMATRA,TANJUNGPUTINGNATIONALPARKINKALIMANTANANDBUNAKENINMANADO,AREPLACESTHEEMBASSYOFFICIALRECOMMENDSFORTOURISTSINDONESIABOASTSWORLDCLASSSURFINGFACILITIESALONGTHESHORESOFSUMATRADOWNTONUSATENGGARAISLANDSRECENTLY,LOMBOK,ASISTERISLANDEASTOFBALIHASGOTTENSOMUCHATTENTIONFROMCHINESETRAVELERSTHATSOMETRAVELAGENCIESAREOFFERINGTWINPACKAGESFORBALIANDLOMBOKTHEBESTTIMETOVISITBALIISFROMMAYTOOCTOBER,WHENITDOESNTRAINMUCHANDTHEWEATHERISCOOLATPRESSTIME,AWEEKLONGPACKAGEFROMSHANGHAIORGUANGZHOUTOBALIWASALITTLEMORETHAN3,000YUAN480ONCTRIPSWEBSITECHINESETOURISTSHAVETAKENASHINETORAFTINGINUBUD,SIGHTSEEINGINTANAHLOT,ANDSURFINGANDDIVINGINNUSALEMBONGAN,SAYSDAICIVETCOFFEE,ESSENTIALOILANDWOODCARVINGSAREAMONGTHEMOSTPOPULARPRODUCTSFORCHINESETOURISTSININDONESIA,SHESAYSINDONESIANFOODISALSOSOMETHINGTHATVISITORSSHOULDNTMISSLOCALCUISINESFEATUREMANYSPICES,INCLUDINGPEPPERSANDCLOVES,ANDC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