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International Monetary FundMoldova and the IMF Press Release:IMF Executive Board Completes Second Review Under the Extended Credit Facility and the Extended Fund Facility Arrangements with Moldova, Approves US$79 Million Disbursement April 7, 2011Country’s Policy Intentions DocumentsE-Mail Notification Subscribe or Modify your subscription Moldova: Letter of Intent, Supplementary Memorandum of Economic and Financial Policies, and Technical Memorandum of UnderstandingMarch 24, 2011M OLDOVA:L ETTER OF I N TE N TChişinău, March 24, 2011 Mr. Dominique Strauss-KahnManaging DirectorInternational Monetary Fund700 19th Street NWWashington, DC 20431 USADear Mr. Strauss-Kahn:The economic program supported by the IMF is playing a crucial role in restoring stability and rebuilding confidence in Moldova. With growth significantly exceeding projections in 2010, GDP has broadly recovered to pre-crisis levels. Inflation is under control, and the fiscal deficit has narrowed substantially. These remarkable results were achieved notwithstanding the challenges that the economy faces: fiscal adjustment and promotion of export-led growth require profound structural reforms; rising international food and fuel prices rekindle inflation pressures; job creation lags behind and unemployment still exceeds pre-crisis levels.The program is broadly on track. All quantitative performance criteria for end-September and most indicative targets for end-December 2010 were observed. However, the difficult political environment of 2010 and unforeseen technical complications have taken their toll, and several structural benchmarks under the program were delayed. In the coming period, we will move expeditiously to implement these measures, as well as the new reforms set forth in our agreement with the IMF. The 2011 fiscal budget consistent with the program objectives will be adopted as a prior action for completion of this review. In addition, we have prepared the Annual Progress Report on the implementation of our National Development Strategy and circulated it to the IMF Executive Board for information.In consideration of our strong record of program implementation, we request the completion of the second review of the program supported by the Extended Credit Facility and the Extended Fund Facility arrangements and the associated disbursement of SDR 50 million. As the Executive Board consideration of our request falls in early April 2011, we also request waivers of applicability of the relevant end-March performance criteria. The third program review, assessing performance based on end-March 2011 performance criteria and relevant structural benchmarks, is envisaged for June 2011. Moldova remains committed to improving the well-being of the population through reforms that promote sustainable growth and reduce poverty. In the period ahead, our program will focus on maintaining the targeted pace of fiscal adjustment; reining in inflation pressures; strengthening financial stability of the banking sector; restructuring the energy sector; rolling out the long-awaitededucation and other structural reforms that would support Moldova’s reorientation toward export-led growth.We believe that the policies set forth in the attached Supplementary Memorandum of Economic and Financial Policies (SMEFP) are adequate to achieve these objectives but will take any additional measures that may become appropriate for this purpose. We will consult with the IMF on the adoption of such additional measures in advance of revisions to the policies contained in the SMEFP, in accordance with the Fund’s policies on such consultation. We will provide the Fund with the information it requests for monitoring progress during program implementation. We will also consult the Fund on our economic policies after the expiration of the arrangement, in line with Fund policies on such consultations, while we have outstanding purchases in the upper credit tranches. Sincerely yours,/s/Vladimir FilatPrime MinisterofRepublicMoldovatheGovernmentof/s/ /s/NegruţaVeaceslavValeriu LazărFinanceofDeputy Prime Minister MinisterEconomyMinisterof/s/Dorin DrăguţanuGovernorNational Bank of MoldovaAttachment: Supplementary Memorandum of Economic and Financial PoliciesUnderstandingofMemorandumTechnicalS UPPLEME N TARY M EMORA N DUM OF E CO N OMIC A N D F I N A N CIAL P OLICIESMarch 24, 20111.The present document supplements and updates the Memoranda of Economic and Financial Policies (MEFPs) signed by the authorities of the Republic of Moldova on January 14, 2010 and June 30, 2010. It accounts for recent macroeconomic developments and introduces policy adjustments, as well as additional policies necessary to achieve the objectives of the program. We remain determined to meeting our commitments made previously under the program.I. M ACROECO N OMIC D EVELOPME N TS A N D O UTLOOK2.Growth outperformed expectations in 2010, and the economic expansion is set to continue. Real GDP rebounded by 6.9 percent in 2010, more than offsetting the economic contraction of 6 percent recorded in 2009. We expect the economic growth to return to its sustainable pace of 4½-5 percent in 2011 and thereafter. Expansion of domestic demand, exports, and investment are expected to drive activity in the near term, with tailwinds from trade liberalization reforms, a more favorable external environment, and improving competitiveness.3.Barring severe external shocks, disinflation should continue in 2011-12. Despite adjustment of energy tariffs, depreciation of the leu, and higher excise rates, inflation remained under control at around 8 percent in 2010, while core inflation declined below 5 percent. Under our baseline assumptions for international food and energy prices, we expect that inflation will decline further to 7½ percent in 2011 and about 5 percent by end-2012, the medium-term target set by the NBM. However, we recognize the risk that further surges in international food and energy prices and faster than expected rebound in domestic demand can temporarily push headline inflation above the projected path.4.Strong economic recovery boosted budget revenues and helped improve the fiscal position. In 2010, revenue significantly exceeded the program projections in nominal terms, but underperformed as percent of GDP, mainly due to high contribution to growth of the largely untaxed agriculture. Expenditure targets were also comfortably met, albeit largely due to under-spending of the capital budget caused by capacity constraints. As a result, the cash budget deficit narrowed to 2½ percent of GDP in 2010, far below the program target of5.4 percent of GDP.5.After a sharp drop to single digits in 2009, the external current account deficit widened in 2010 and will remain elevated in 2011. Rising demand for consumer and investment goods has pushed the current account deficit to an estimated 12¾ percent of GDP in 2010. The same demand factors, along with higher costs of energy imports, will likely propel the deficit even higher in 2011. The elevated deficit in 2011 will be largely financed by official assistance, private capital flows, and FDI. As the economy’s borrowing space is filling up quickly, we realize that further external borrowing should proceed at a more measured pace. We expect that from 2013, thanks to our exportpromotion efforts and economic recovery in trading partners, higher exports will more than offset the rise in imports, and the current account deficit would decline towards 10 percent of GDP.6.The situation in the financial sector has improved as well, with domestic credit rebounding and nonperforming loans declining. After the decline of 2009, domestic bank credit expanded by about 13 percent in 2010, and interest rates have declined. Meanwhile, the share of nonperforming loans declined to 13.3 percent, in part reflecting write-offs. Moreover, banks maintain large liquidity and capital buffers, remaining resilient to potential risks.II. R EVISED P OLICY F RAMEWORK FOR 2011-12A. Fiscal Policy7.Building on the better-than-expected fiscal outcome in 2010, the structural fiscal adjustment will stay on course in 2011-12. Our goal is to bring down the structural fiscal deficit excluding grants—the fiscal deficit adjusted for the effects of economic cycles—from 5½ percent of GDP at end-2010 through 4½ percent of GDP in 2011 to 3½ percent of GDP by 2012. This would largely rid the budget from its dependency on exceptional foreign aid and make public finances more resilient to macroeconomic risks. In this context, we will continue to contain the unaffordable public sector wage bill and low priority current spending, while strengthening revenue through selected tax policy measures and improved tax administration. Using the created fiscal space to increase infrastructure investment and provide well-targeted social assistance to the most vulnerable will allow us to achieve our broader development goals.8.As a next step, we will adopt a 2011 budget with a deficit of 1.9 percent of GDP as a prior action. We project that the budget revenue will amount to 37¾ percent of GDP in 2011, on account of continued progress in the tax administration reform, increased excise rates on tobacco and hard liquor—in line with our EU Association agenda—and updates of selected local taxes and fees. Implementation of various structural reforms, described below, will allow us to reduce current expenditure by 1½ percent of GDP to 34½ percent of GDP. At the same time, priority social assistance spending will be safeguarded, and capital expenditure will increase to 5¼ percent of GDP. We will seek to maintain the targeted structural fiscal adjustment in case the economic outlook and budget revenue deviate from our current projections.9.With immediate fiscal pressures easing, structural reforms will help contain the large public sector wage bill while creating space for poverty reduction actions. The significant optimization efforts in the education sector (¶19) will help finance the increase of teachers’ wages planned for September 2011. During 2011, other public wage restraints will remain in place as described in Law 355, as amended in October 2009. The only exception will be made for low-income auxilliary personnel in the budget sector (with salaries below MDL 1500), whose wages will be indexed by 8.5 percent on average from July 1, 2011 to alleviate the impact of higher than expected food and fuel prices and to avoid disincentives to labor market participation. Moreover, public sectoremployment will be capped at 212,000 positions by end-2011, reflecting the effects of the education reforms, while all vacant positions in excess of that level will be eliminated in 2011.10.Greater emphasis will be placed on synchronizing fiscal consolidation efforts at the central and local levels. The local governments will be granted greater control over local tax rates and fees to allow better revenue planning. In particular, by end-March 2011, we will ensure parliamentary passage of the necessary legal amendments to remove ceilings on existing local taxes and fees. This would allow the Chişinău municipality to raise at least MDL 100 million in additional revenues to finance, among other things (discussed in ¶21), its program of granting wage supplements and heating assistance in 2011. The practice of granting these payments will be discontinued at end-2011. The Ministry of Finance will verify compliance with these commitments.11.Going forward, we will continue trimming down current spending while creating sufficient space for the large public investment needs. In 2012, we aim to reduce the budget deficit further to ¾ percent of GDP, mainly through further rationalization of current spending (1 percent of GDP), sustained by structural reforms (¶¶19-22) that will commence in 2011 and bear fruit over the medium term. Ensuring sustainability of public finances in the medium term will also require implementation of the following measures:∙To reduce spending on goods and services, we will persevere with our procurement reform, assisted by the World Bank. The reform, to be phased in during 2011, will lower the budget costs by automating the bids for delivery of goods and services in the government’scentralized procurement agency.∙To improve control over budget planning and execution, we have drafted a law on public finance and accountability which will introduce a rule-based fiscal framework, enhance fiscal discipline, and improve transparency. We expect the law to be passed by Parliament by end-September 2011 and used in the preparation of the 2012 budget.∙To ensure the most effective allocation of capital expenditure, we will review the list of existing and envisaged capital projects, with a view to prioritize execution on the basis oftheir viability and economic growth potential. The review will also take into account pastexecution rates and capacity for implementation.∙To ensure implementation of the recently approved tax compliance strategy, by April 30, 2011, the State Tax Service (STS) will put in place operational plans for the strategyimplementation, including audit, collection of arrears, and taxpayer service activities(structural benchmark). In addition, by September 30, 2011, we will draft and submit toParliament legislation to allow indirect assessment of individuals’ income based on theirassets and other indicators as specified in the compliance strategy. On this basis, byDecember 31, 2011, we will prepare operational plans to strengthen audit, enforcement,outreach to, and education of high-wealth individuals regarding their tax compliance.∙We will reform the outdated mechanism for sick leave benefits. By March 31, 2011, we will amend legislation to assign the financial responsibility for the first day of sick leave to theemployee and the second day to the employer, effective July 1, 2011 (structural benchmark for end-April). Further legal amendments—to accompany the passage of the 2012 budget—will increase the number of sick leave days covered by employers to 3 in 2012, 4 in 2013, and6 in 2014.∙Early retirement privileges will be gradually phased out. By March 31, 2011, we will adopt legislation that, starting July 1, 2011, would raise the statutory retirement age of civilservants, judges, and prosecutors by six months every year until it reaches the regularretirement age (structural benchmark for end-April). This legislation will also extend the requirement to pay social contributions to all persons employed in Moldova in line withbilateral treaties. Another related piece of legislation, also to be passed by March 31, 2011,will put in place a policy of increasing the years of contribution required for full pensioneligibility from 30 to 35 years (and from 20 to 25 years for military and police personnel), by6 months every year, starting July 1, 2011.∙Building on the findings and recommendations of the recent IMF TA mission, we will implement measures to rationalize the use of health care. In particular, from January 1, 2012 we will introduce a copayment of 20 lei for primary care visits for uninsured patients, tomotivate them to enroll into the health insurance system. From January 1, 2013, we willintroduce small copayments for each doctor and hospital visit (5 lei for primary care, 10 leifor specialists, and 20 lei for hospital admissions) for all other categories of patients,including those who currently receive medical services free of charge. This policy will raise revenue and deter the use of unnecessary care, thus reducing the burden on the system. Tothis end, by end-April 2011 we will prepare an action plan detailing needed legislativechanges, technical preparations, and public information campaign.B. Monetary and Exchange Rate Policies12.The N BM’s monetary policy will be focused on achieving its end-2012 inflation objective of 5 ± 1½ percent. Given the fast economic recovery, closing output gap, and inflation pressures from rising international food and energy prices, the NBM’s monetary policy stance will gradually shift from supporting the recovery to addressing inflation risks. Specifically, it should focus on anchoring expectations—thereby countering the second-round effects from surging food and energy prices—and preventing excessive credit expansion. In this context, the NBM’s recent tightening measures—the 100 basis points hike in the policy interest rate and the increase in required reserve ratio from 8 percent to 11 percent— adequately address current inflation concerns. Further tightening should be conditional on marked acceleration of credit growth or rising inflation expectations.13.At the same time, the N BM will continue to strengthen the operational and legal aspects of its monetary policy framework. Consistent with the transition to inflation targeting, theindicative target for reserve money under the program will be discontinued after March 2011. Nevertheless, the NBM will continue to monitor money growth closely as an indicator of the state of domestic demand and sharp sustained moves may warrant policy action. In parallel, the NBM will continue to further enhance its communication, research, and forecasting capacities. As regards the legal framework, by end-September 2011, the NBM will propose amendments to the central bank law to strengthen its independence in line with the international best practice and establish appropriate mechanisms of internal control over NBM’s corporate governance.14.Alongside, the N BM’s exchange rate policies will remain consistent with program objectives. Specifically, NBM interventions in the foreign exchange market will continue to aim at smoothing erratic movements, but not resist sustained depreciation pressures. Should capital inflows exceed program projections, the NBM will accelerate the pace of reserve accumulation to ensure adequate buffers against the still high external vulnerabilities.C. Financial Sector Policy15.To strengthen financial stability, we will address the quasi-fiscal liabilities stemming from recent crisis management efforts. The Government’s decision to shield from losses the depositors of Investprivatbank (IPB) that failed in 2009 was a necessary step to avoid potential panic and deposit runs. However, paying out these deposits by means of a loan from the majority state-owned Banca de Economii (BEM) to IPB—in turn, enabled by a liquidity-providing loan from the NBM—has created a burden on BEM’s balance sheet that is now inhibiting its development. To address this problem, by end-May 2011 the Government will issue to BEM a long-term bond equal to the residual face value of BEM’s loan to IPB by either purchasing this loan or—subject to agreement of BEM’s minority shareholders—recapitalizing the bank. Meanwhile, the NBM will consider a limited extension of its loan to BEM to mitigate the attendant liquidity risk, and will work with BEM and the IPB liquidator to accelerate the sale of IPB assets. The Deposit Guarantee Fund will assume the responsibility for the net cost of the payout to IPB depositors and may introduce an extraordinary deposit insurance premium to gradually reimburse the Government for the cost of the bond issued to BEM.16.To handle future risks better, we aim to put in place the remaining elements of our contingency planning framework. Recent strengthening of the bank resolution framework and the establishment of a high-level Financial Stability Committee (FSC) were followed by signing of a memorandum of understanding (MoU) between key institutions involved in responding to financial emergencies. As a next step, we aim to put in place specific contingency plans for each MoU participant by end-June 2011. These plans will establish a contingency framework based on a clear set of instruments, division of roles, responsibilities, as well as coordination channels between the involved parties.17.Looking ahead, as credit growth picks up speed, the N BM will need to strengthen its bank supervision framework by improving data collection and reducing scope for regulatoryarbitrage. To this end, the NBM, based on best international practices, will develop a new reporting system for commercial banks allowing a more detailed analysis of financial sector data. In addition, by end-September 2011, the NBM and the National Commission for Financial Markets, with assistance from the World Bank, will explore options and make proposals to consolidate all credit institutions—including banks, leasing companies, savings and credit associations, and microfinance institutions—as well as insurance companies and pension funds under a common supervisory framework. Finally, by end-September 2011, the NBM in cooperation with the World Bank will evaluate the feasibility of establishing a public credit bureau to promote information exchange and prudent lending policies by banks.18.Despite earlier delays, measures to strengthen the debt restructuring and contract enforcement frameworks are being developed and will be implemented in the coming months. The NBM has already allowed faster reclassification of restructured loans into lower-risk categories. We will now ensure by end-September 2011 parliamentary passage of the legal amendments described in the SMEFP of June 30, 2010 (¶15), to enhance the speed and predictability of collateral execution by banks and to strengthen incentives for banks to restructure nonperforming loans (structural benchmark). Furthermore, with technical assistance from the World Bank and in consultation with the IMF staff, we will seek to strengthen and simplify other aspects of the insolvency framework. Specific draft legal amendments in this area will be adopted by the Government by March 2012.D. Structural ReformsRaising Efficiency of the Public Sector19.In the coming months, we will roll out the comprehensive reform of the oversized education sector. Its main goals are to eliminate excess capacity, create a leaner and better-equipped education system with adequately trained and paid staff, and provide education that meets demands of the modern economy. The reform will seek class, school, and employment consolidation. A large part of the eventual budget savings and financial assistance from the World Bank will be used to improve school quality, secure transportation for students, and repair school bus routes. Nevertheless, the reform will save about 0.5 percent of GDP on a net permanent basis from 2013 on. Our reform strategy is based on the following elements:∙Class size optimization. By September 1, 2012, we will increase class size to 30-35 students in large schools and 25-30 students in the rest. For this purpose, we will pass legalamendments to eliminate the existing norms prescribed in the Law on Education by end-July 2011. This would reduce the number of teaching positions by 1,736, including 390 positions in 2011, and lead to estimated annual savings of about MDL 94 million.∙Optimization of the school network. Gradual consolidation of the school network through closure of schools with low enrollment and securing transportation of students to nearby“hub” schools will commence this year. Its full implementation during 2011-13 would reducethe number of teaching and non-teaching positions by 2,661 and 1,426 respectively and, when completed, will generate savings of about MDL 136 million a year. We will aim to limit the attendant transportation costs to MDL 61 million per year, and will seek grant assistance from the international financial community to defray this cost.∙Reduction of non-teaching personnel and vacant positions. As a first step, we will immediately freeze hiring of non-teaching staff and eliminate 2,400 vacant positions in thesector. Alongside, we will include in the budget law for 2011 a provision establishing wage bill ceiling for education sector, resulting in all rayons reducing personnel in educationinstitutions on average by 5 percent from their level of end 2010 (5,300 positions nationwide) before academic year 2011/12. These measures would provide savings of MDL 175 million on a full-year basis.∙Increasing flexibility of labor relations in the sector. Local authorities also need support and more flexibility to be able to consolidate schools and classes. By end-July 2011, we willadopt legal amendments to the Labor Code and other enabling legislation to (i) make fixed-term (one year) contracts mandatory for teachers beyond retirement age; and (ii) allow school principals’ hiring and dismissal decisions to be based on business need and performancerather than tenure. Estimated annual savings from this measure amount to MDL 48 million. ∙Rollout of a per-student financing system. Following successful implementation of per-student financing in the pilot rayons of Cauşeni and Rişcani, the system will be expandedstarting January 1, 2012 to 9 additional rayons, as well as municipalities of Chişinău andBalţi. The system will create strong incentives to optimize schools’ financial performance. Its nationwide implementation will take place in 2013.∙Putting social protection costs in education on a means-tested basis. By end-June 2011, in consultation with the World Bank and other partners, we will conduct a thorough review ofall social expenditure in the education budget (scholarships, dormitory assistance, schoolmeals, etc.) to explore options for better targeting of such assistance to the most vulnerablegroups.In consultation with the World Bank, the Government will develop and, by end-March 2011, adopt a detailed action plan to implement this reform.20.We will reform the civil service in a way that increases efficiency without destabilizing the fiscal position. To this end, we have developed descriptions of new job functions and responsibilities for staff in central government administration along with a merit- and performance-based wage system for civil servants. Implementation of this reform will start in October 2011, and will ensure that the reform does not affect the aggregate public sector wage bill as a ratio to GDP. 21.As regards the energy sector, we will strive to achieve a stable framework for payments of current bills, pending a comprehensive sector restructuring strategy to be finalized and implemented in cooperation with the World Bank and other partners. To ensure a stablefunctioning of the sector, the Ministry of Economy, the Chişinău municipality authorities, and the key participants in the energy sector will seek to negotiate in good faith a MoU with the following key elements: (i) a monthly schedule of payments to energy suppliers that is consistent with typical collection lags in Termocom’s receivables during the heating season, (ii) full repayment of current arrears by Termocom before the following heating season; (iii) a mechanism for covering the cash gap arising from collection lags in Termocom or a bank guarantee from the Chişinău municipality backing Termocom’s adherence to the agreed payment schedule; (iv) creditors’ commitment to abstain from blocking bank accounts as long as the MoU is observed. In this context, the Chişinău municipality will budget for and pay in full its remaining debt to Termocom of MDL 64 million by end-March 2011.22.Meanwhile, we will adopt a number of legal and regulatory amendments which would help ensure cost recovery in the heating sector. By end-August 2011, we will adopt the necessary legal and/or regulatory amendments to raise the heating fee for apartments disconnected from central heating from 5 percent to 20 percent of the average heating bill. This increase is in line with regional practices and would mostly affect consumers with relatively high incomes. At the same time, the Ministry of Regional Development and Construction, the Chişinău municipality, Termocom, and the water distributor Apă Canal will seek to put an end to persistent losses caused by under-billing for hot and cold water delivery; other municipalities will seek to resolve this issue as well. And to facilitate timely collection of heating bills, by end-August 2011, we will adopt the necessary legal and/or regulatory amendments introducing a minimum payment of 40 percent of the monthly bill and setting August 1 as the deadline for settling all heating bills for the past heating season.23.With the international investment climate gradually improving, the government will accelerate the efforts to divest its noncore assets. In the first half of 2011 the government, with assistance from IFC, will put in place an advisor to review various options for private sector participation in Moldtelecom. At the same time, by mid-2011, the government will expand the list of state assets subject to privatization. This will pave the way for privatization of other large public companies. By end-September 2011, the government will approach various international financial institutions, seeking an advisor to explore options to divest Air Moldova as soon as possible. Also by end-September 2011, we shall develop a roadmap for the privatization of Banca de Economii, and, if need be, resume the engagement of the privatization advisor.Improving the Business Environment and Removing Barriers for Trade24.The wheat export ban introduced in response to dwindling grain stocks in early 2011 will be abolished as soon as possible, and we will not introduce any new barriers to trade. We plan to abolish this ban by end-April 2011, provided that domestic and regional grain shortages are alleviated. Moreover, we shall refrain from introducing any new tariff or non-tariff barriers to exports. In addition, by end-May 2011 we will conduct an assessment of the existing tariff and non-tariff barriers to trade and their consistency with Moldova’s WTO commitments with regard to market access, and will develop roadmap for their gradual elimination.。
第一章 初等函数习题答案练习题1.11. 数3.1415926是有理数。
2.是无理数。
3. 数4. 有限区间有4个，分别为(),a b [],a b [),a b (],a b 无限区间有5个，分别为(],b -∞ (),b -∞(),a +∞ [),a +∞ (),-∞+∞5. 实数集合{}||2|1,x x x R -<∈用区间表示为()1,36. 实数集合{}||1|2,x x x R -<∈可以认为是1为中心，长度为4的开区间。
7. 实数集合{}||1|2,x x x R -<∈可以称为1的邻域。
8. 以点3为中心，区间长度为1的邻域表示为{}||3|1,x x x R -<∈ 练习题1.21. 函数的三种表示法分别为公式法，图像法，列表法。
2. 单调增函数的是y x =，3y x =，xy e =，ln y x =，lg y x =，tan y x =，arcsin y x = arctan y x = ；单调减函数的是xy e -=， cot y x = 分区间的增减函数是2y x =， sin y x =，cos y x =. 3. 函数2()ln f x x =和()2ln g x x =不是相同函数。
由于2()ln f x x =的定义域是0x ≠；()2ln g x x =的定义域是0x >。
4. 函数()f x x =和()g x =不是相同函数。
由于()f x x =的值域是()f x R ∈，()g x 的值域是()0g x ≥。
5. 求下列函数的定义域：（1）y =解：[)(]1,,1+∞⋃-∞-（2）21()1f x x =- 解：[)2,-+∞且1x ≠± （3）()ln(1)f x x =+ 解：()1,-+∞（4）()lg(1)f x x =- 解：(][),22,-∞-⋃+∞6. 判断下列函数的奇偶性：（1）33y x x =+ 解：奇函数。
微积分第三版上册课后练习题含答案微积分是数学的一个分支，它主要研究函数、极限、连续、导数、积分等概念和它们之间的关系。
微积分是自然科学、工程技术和经济管理等领域中不可或缺的数学工具。
本文将介绍微积分第三版上册的课后练习题，以及它们的答案和解析。
章节列表微积分第三版上册共分为12章，分别是：1.函数与极限2.导数及其应用3.曲线图形的相关概念4.定积分5.定积分应用6.不定积分7.不定积分的应用8.微分方程初步9.空间解析几何10.空间直线与平面11.空间曲面12.重积分每一章都包含了大量的练习题，这些题目是对每个章节中理论知识点的考察和巩固，同时也能够帮助读者构建更深入的理解。
练习题样例下面是微积分第三版上册第一章的一组练习题样例：1.1节练习1.求函数$f(x)=\\frac{x-1}{x+1}$在点x0=2处的导数。
2.求极限$\\displaystyle\\lim_{x \\to +\\infty}(\\sqrt{x^2+3x}-\\sqrt{x^2-5})$。
3.求函数$f(x)=\\sqrt{1+x}-1$的二阶导数。
1.2节练习1.求$f(x)=\\frac{1}{x}$的导函数和导数。
2.已知函数f(x)=x3+3x2+1，求它的单调区间和极值点。
3.求函数f(x)=x4−8x2的导函数和导数。
课后练习题答案微积分第三版上册的课后练习题答案可以在教材的补充练习答案中找到，答案涵盖了书中各章节的所有练习题。
下面是上述练习题的答案和解析。
1.1节练习答案1.$f'(2)=\\frac{2}{9}$2.$\\displaystyle\\lim_{x \\to +\\infty}(\\sqrt{x^2+3x}-\\sqrt{x^2-5})=+\\infty$3.$f''(x)=\\frac{1}{4(x+1)^{\\frac{3}{2}}}$1.2节练习答案1.$f'(x)=-\\frac{1}{x^2}$，$f''(x)=\\frac{2}{x^3}$2.f(x)在$(-\\infty,-1)$上单调递减，在$(-1,+\\infty)$上单调递增。
微积分课后题答案习题详解IMB standardization office【IMB 5AB- IMBK 08- IMB 2C】第二章习题2-11. 试利用本节定义5后面的注（3）证明：若lim n →∞x n =a ,则对任何自然数k ，有lim n →∞x n +k =a .证：由lim n n x a →∞=，知0ε∀>，1N ∃，当1n N >时，有取1N N k =-，有0ε∀>，N ∃，设n N >时（此时1n k N +>）有 由数列极限的定义得 lim n k x x a +→∞=.2. 试利用不等式A B A B -≤-说明：若lim n →∞x n =a ,则lim n →∞∣x n ∣=|a|.考察数列x n =(-1)n ，说明上述结论反之不成立.证：而 n n x a x a -≤- 于是0ε∀>，,使当时，有N n N ∃>n n x a x a ε-≤-< 即 n x a ε-<由数列极限的定义得 lim n n x a →∞=考察数列 (1)nn x =-，知lim n n x →∞不存在，而1n x =，lim 1n n x →∞=，所以前面所证结论反之不成立。
3. 利用夹逼定理证明：(1) lim n →∞222111(1)(2)n n n ⎛⎫+++ ⎪+⎝⎭=0； (2) lim n →∞2!n n =0.证：（1）因为222222111112(1)(2)n n n n n n n n n n++≤+++≤≤=+ 而且 21lim0n n →∞=，2lim 0n n→∞=， 所以由夹逼定理，得222111lim 0(1)(2)n n n n →∞⎛⎫+++= ⎪+⎝⎭. （2）因为22222240!1231n n n n n<=<-，而且4lim 0n n →∞=，所以，由夹逼定理得4. 利用单调有界数列收敛准则证明下列数列的极限存在.(1) x n =11n e +，n =1,2,…；(2) x 1x n +1,n =1,2,…. 证：（1）略。
习题1—1解答1． 设y x xy y x f +=),(，求),(1),,(),1,1(),,(y x f y x xy f y x f y x f -- 解yxxy y x f +=--),(；x xy y y x f y x y x xy f x y xy y x f +=+=+=222),(1;),(;1)1,1(2． 设y x y x f ln ln ),(=，证明：),(),(),(),(),(v y f u y f v x f u x f uv xy f +++=),(),(),(),(ln ln ln ln ln ln ln ln )ln )(ln ln (ln )ln()ln(),(v y f u y f v x f u x f v y u y v x u x v u y x uv xy uv xy f +++=⋅+⋅+⋅+⋅=++=⋅=3． 求下列函数的定义域，并画出定义域的图形： （1）;11),(22-+-=y x y x f（2）;)1ln(4),(222y x y x y x f ---=（3）;1),(222222cz b y a x y x f ---=（4）.1),,(222zy x z y x z y x f ---++=解（1）}1,1),{(≥≤=y x y x D（2）{y y x y x D ,10),(22<+<=（3）⎫⎩⎨⎧++=),(22222b y a x y xD（4）{}1,0,0,0),,(222<++≥≥≥=z y x z y x z y x D4．求下列各极限： （1）22101limy x xy y x +-→→=11001=+- （2）2ln 01)1ln(ln(lim022)01=++=++→→e yx e x y y x（3）41)42()42)(42(lim 42lim000-=+++++-=+-→→→→xy xy xy xy xy xy y x y x（4）2)sin(lim )sin(lim202=⋅=→→→→x xy xy y xy y x y x5．证明下列极限不存在：（1）;lim 00yx y x y x -+→→ （2）2222200)(lim y x y x y x y x -+→→ （1）证明 如果动点),(y x P 沿x y 2=趋向)0,0( 则322lim lim0020-=-+=-+→→=→x x xx y x y x x x y x ；如果动点),(y x P 沿y x 2=趋向)0,0(，则33lim lim0020==-+→→=→y yy x y x y y x yx所以极限不存在。
微 积 分 课 后 习 题 答 案习 题 一 （A ）1．解下列不等式，并用区间表示不等式的解集：（1）74<-x ； （2）321<-≤x ；（3）)0(><-εεa x ； （4）)0,(0><-δδa x ax ；（5）062>--x x ；（6）022≤-+x x ．解：1)由题意去掉绝对值符号可得：747<-<-x ，可解得j .113．x <<-即)11,3(-. 2）由题意去掉绝对值符号可得123-≤-<-x 或321<-≤x ，可解得11≤<-x ,53<≤x .即]5,3[)1，1(⋃-3）由题意去掉绝对值符号可得εε<-<-x ,解得εε+<<-a x a .即)a , (εε+-a ；4）由题意去掉绝对值符号可得δδ<-<-0x ax ,解得ax x ax δδ+<<-00，即ax a x δδ+-00 , () 5）由题意原不等式可化为0)2)(3(>+-x x ,3>x 或2-<x 即）（3， 2） ， (∞+⋃--∞. 6）由题意原不等式可化为0)1)(2(≤-+x x ,解得12≤≤-x .既1] , 2[-.２．判断下列各对函数是否相同，说明理由： （1）x y =与x y lg 10=； （2）xy 2cos 1+=与x cos 2；（3）)sin (arcsin x y =与x y =；（4）)arctan (tan x y =与x y =；（5）)1lg(2-=x y 与)1lg()1lg(-++=x x y ； （6）xxy +-=11lg 与)1lg()1lg(x x x +--=．解：1)不同，因前者的定义域为) , (∞+-∞，后者的定义域为) , 0(∞+； 2)不同,因为当))(2 , )212((ππ23k k x k ++∈+∞-∞- 时，02cos 1 >+x ,而0cos 2<x ；3）不同，因为只有在]2, 2[ππ-上成立； 4）相同；5)不同，因前者的定义域为) , (11) , (∞+⋃--∞)，后者的定义域为) , 1(∞+； 6）相同3．求下列函数的定义域（用区间表示）： （1）1)4lg(--=x x y ； （2）45lg 2x x y -=；（3）xx y +-=11； （4）)5lg(312x x x y -+-+-=； （5）342+-=x x y ；（6）xy xlg 1131--=；（7）xy x-+=1 lg arccos 21； （8）6712arccos2---=x x x y ．解：1)原函数若想有意义必须满足01>-x 和04>-x 可解得 ⎩⎨⎧<<-<41 1x x ,即)4 , 1()1 , (⋃--∞.2)原函数若想有意义必须满足0452>-x x ,可解得 50<<x ,即)5 , 0(.3)原函数若想有意义必须满足011≥+-xx,可解得 11≤<-x ,即)1 , 1(-. 4)原函数若想有意义必须满足⎪⎩⎪⎨⎧>-≠-≥-050302x x x ,可解得 ⎩⎨⎧<<<≤5332x x ,即) 5 , 3 (] 3 , 2 [⋃,3]. 5)原函数若想有意义必须满足⎪⎩⎪⎨⎧≥--≥+-0)1)(3(0342x x x x ,可解得 ⎩⎨⎧≥-≤31x x ,即(][) , 3 1 , ∞+⋃-∞.6)原函数若想有意义必须满足⎪⎩⎪⎨⎧≠-≠>0lg 100x x x ,可解得⎩⎨⎧><<10100x x ,即) , 10()10 , 0(∞+⋃. 7)原函数若想有意义必须满足01012≤≤-x 可解得21010--≤<x 即]101 , 0()0 , 101[22--⋃- 8)原函数若想有意义必须满足062>--x x ,1712≤-x 可解得) 4 , 3 (] 2 , 3 [⋃--.4．求下列分段函数的定义域及指定的函数值，并画出它们的图形： （1）⎪⎩⎪⎨⎧<≤-<-=43,13,922x x x x y ，求)3( , )0(y y ；（2）⎪⎪⎩⎪⎪⎨⎧∞<<+-≤≤-<=x x x x x x y 1, 1210,30,1，求)5( , )0( , )3(y y y -．解：1）原函数定义域为：)4 , 4(-3)0(==y 8)3(==y .图略2)原函数定义域为：) , (∞+-∞31)3(-=-y 3)0(-==y 9)5(-=y y(5)＝-9.图略5．利用x y sin =的图形，画出下列函数的图形：(1)1sin +=x y ； （2）x y sin 2=； （3）⎪⎭⎫⎝⎛+=6sin πx y ．解：x y sin =的图形如下(1)1sin +=x y 的图形是将x y sin =的图形沿沿y 轴向上平移1个单位(2)x y sin 2=是将x y sin =的值域扩大2倍。
习题一答案(A)1.1. 求下列函数的定义域：(1) 22-+=x x y ; (2) )sin(x y =;(3) 2)1lg(--=x x y ; (4) 22114xx y -+-=; (5) x xx y -++-=11lg21)1arcsin(; (6) ⎩⎨⎧><+=)0(ln )0(12x xx x y . (1)解：022≥-+x x21-≤≥x x 或∴定义域为),1[]2,(+∞--∞ .(2)解：⎩⎨⎧≥≥00)sin(x xπππ+≤≤k x k 22∴定义域为{},1,0,)12(42222=+≤≤k k x k x ππ.(3) 解：⎩⎨⎧≠->-0201x x21≠>x x 且∴定义域为),2()2,1(+∞ .(4)解： ⎩⎨⎧≠-≥-010422x x ⎩⎨⎧±≠≤≤-122x x ∴定义域为]2,1()1,1()1,2[ ---.(5) 解：⎪⎪⎩⎪⎪⎨⎧≠->-+≤-≤-01011111x xxx ⇒ ⎪⎩⎪⎨⎧≠<<-≤≤11120x x x ∴定义域为)1,0[.(6) 解：定义域为),0()0,(+∞-∞ .2已知23)(2-+=x x x f ,求)1(,1),(),1(),1(),0(+⎪⎭⎫⎝⎛--x f x f x f f f f . 解：2200)0(2-=-+=f2231)1(2=-+=f 423)1()1(2-=---=-f232)(3)()(22--=--+-=-x x x x x f231)1(2-+=xx x f 252)1(3)1()1(22++=-+++=+x x x x x f3. 已知⎩⎨⎧≥<+=1ln 113)(x x x x x f ,求)2(),1(),0(f f f .解：1103)0(=+⨯=f01ln )1(==f 2ln )2(=f4. 讨论下列函数的单调性（指出其单调增加区间和单调减少区间） (1) x x y ln +=; (2) xe y =; (3) 24x x -. 解：（1）定义域为),0(+∞，设210x x <<，0)ln (ln ln ln 1212112212>-+-=--+=-x x x x x x x x y y故在定义域内为单调增函数，单调增加区间为),0(+∞. (2) 定义域为实数R,当021<<x x 时，21x x >，021>-x x e e ，函数为减函数； 当210x x <<时，21x x <，021<-x x ee，函数为增函数.故单调减少区间为)0,(-∞,单调增加区间为),0(+∞. (3) 定义域为[]4,0，4)2(422+--=-=x x x y当20≤≤x 时，2)2(--x 为增函数，4)2(2+--x 也为增函数，当42≤≤x 时，2)2(--x 为减函数，4)2(2+--x 也为减函数.故单调增加区间为]2,0[,单调减少区间为]4,2[.5. 判别下列函数中哪些是奇函数,哪些是偶函数,哪些是非奇非偶函数. （1）2x ey -=; (2）x x y sin 2=;（3）242x x y -=; （4）2x x y -=;（5）x x y cos sin -=; （6）x xy +-=11lg; （7）)1ln(2x x y -+=; （8）x xx y cos sin +=;（9）x x xx e e e e y ---+=; （10）⎩⎨⎧≥+<-=0101x xx x y .解：（1）定义域为实数R,)()(22)(x y e e x y x x ===----，故函数为偶函数.(2）定义域为实数R,)(sin )sin()()(22x y x x x x x y -=-=--=-，故为奇函数.（3）定义域为实数R,)(2)(2)()(2424x y x x x x x y =-=---=-，故函数为偶函数.（4）定义域为实数R,函数2x x y -=为非奇非偶函数. （5）非奇非偶函数 （6）定义域为011>+-xx，0)1)(1(>+-x x ，即11<<-x ， 0111lg 11lg )()(==+-+-+=+-lg xxx x x y x y ，即)()(x y x -=-y ，故函数为奇函数. （7）定义域为实数R,01ln )1ln()1ln()()(22==-+++=+-x x x x x y x y ，)()(x y x -=-y ，故函数为奇函数.（8）定义域为),0()0,(+∞-∞ ，)(cos sin )cos()sin()(x y x xx x x x x y =+=-+--=-，故函数为偶函数. （9）定义域为),0()0,(+∞-∞ ，)()(x y ee e e e e e e x y xx xx x x x x -=-+-=-+=-----，故函数为奇函数. （10）)(01010101)(x y x xx x x x x x x y =⎩⎨⎧>+≤-=⎩⎨⎧≥--<-+=-，故函数为偶函数.6. 设)(x f 在),(+∞-∞内有定义,证明：)()(x f x f -+为偶函数,而)()(x f x f --为奇函数.证明：令)()()(x f x f x g -+=，)()()(x f x f x h --=，)()()()(x g x f x f x g =+-=-，)(x g 为偶函数， )()()()(x h x f x f x h -=--=-，)(x h 为奇函数.7. 判断下列函数是否为周期函数,如果是周期函数,求其周期： （1）x x y cos sin +=; （2）x x y cos =; （3）)32sin(+=x y ; （4）x y 2sin =; （5）x y 2sin 1+=; （6）xy 1cos =. 解：（1）)4sin(2)cos 22sin 22(2π+=+=x x x y故函数周期为π2.（2）无周期 （3）周期为ππ==22T（4）22cos 1sin 2xx y -==，周期为ππ==22T（5）设)22sin(1)(2sin 12sin 1T x T x x y ++=++=+= , 解得π=T 2 ，2/π=T .（6）无周期8. 讨论下列函数是否有界：（1）221xx y +=; （2）2x e y -=; （3）x y 1sin=; （4）x y -=11; （5）xx y 1cos =.解：（1）1122≤+=xx y ，故函数有界.（2）02≥x ,02≤-x ,102≤<-x e ,故函数有界.（3）11sin≤x，函数有界. （4）xy -=11无界. （5）xx y 1cos =无界.9. 设21)(x x x f -=,求)(cos x f .解：x x x x x f cos sin cos 1cos )(cos 2=-=10. 已知⎩⎨⎧>-≤+=0102)(2x x x x x f ,求)1(+x f 及)()(x f x f -+.解：⎩⎨⎧->-≤++=⎩⎨⎧>+-+≤+++=+1132011)1(012)1()1(22x xx x x x x x x x f⎩⎨⎧<--≥+=-0102)(2x x x x x f ⎩⎨⎧>-≤+=0102)(2x x x x x f⎪⎩⎪⎨⎧>++=<+-=-+01041)()(22x x x x x x x x f x f 11. 已知x x x f -=3)(,x x 2sin )(=ϕ,求)]([x f ϕ,)]([x f ϕ. 解：x x x f 2sin )2(sin )]([3-=ϕ，)(2sin )]([3x x x f -=ϕ 12. (1) 已知 2211xx x x f +=⎪⎭⎫ ⎝⎛+,求)(x f .(2)已知2ln )1(222-=-x x x f ,且x x f ln )]([=ϕ,求)(x ϕ.解：(1) 2)1(12-+=⎪⎭⎫ ⎝⎛+xx x x f ,2)(2-=∴x x f (2)令12-=x t ，11ln)(-+=t t t f ，xx x x f ln 1)(1)(ln ))((=-+=ϕϕϕ，x x x x =-+=-+1)(211)(1)(ϕϕϕ11112)(-+=+-=x x x x ϕ13. 在下列各题中,求由给定函数复合而成的复合函数,并确定定义域： （1）21,x u u y +==； （2）2,ln ,4xv v u u y ===； （3）x v v u u y 21,sin ,3+===;（4）222,tan ,arctan x a v v u u y +===. 解：（1）21x y +=，),(+∞-∞∈x （2）2ln4x y =,由02>x，),0(+∞∈x（3）)21(sin 3x y +=，),(+∞-∞∈x（4）)](arctan[tan 222x a y +=，由2/)(22ππ+≠+k x a ，有⎭⎬⎫⎩⎨⎧∈-+≠∈Z R k a k x x x ,2,22ππ14. 指出下列各函数是由哪些简单函数复合而成的？ （1）x y alog =; （2）x e y -=arctan ;（3）x y 2sin ln =; （4）⎪⎭⎫⎝⎛-=2212arcsin x xy .解： （1）x y alog =，x u = （2）u y arctan =，v e u =，x v -=（3）u y ln =，2v u =，x v sin = （4）2u y =，v u arcsin =，212x x v -= 15. 求下列反函数及反函数的定义域：（1）)31ln(x y -=,)0,(-∞=f D ； （2）29x y -=,]3,0[=f D ;（3）22-+=x x y ,),2()2,(+∞-∞= f D ; （4）2xx e e y --=,),(+∞-∞=f D ;（5）⎩⎨⎧≤<--≤<-=21)2(210122x x x x y . 解：（1）由)31ln(x y -=解得3/)1(ye x -=,故)1(31x e y -=,),0(1+∞=-f D （2）由29x y -=解得29y x -=，故29x y -=,]3,0[1=-f D（3）由22-+=x x y 解得1)1(2-+=y y x ,故1)1(2-+=x x y ,),1()1,(1+∞-∞=- f D （4）由2x x e e y --=同乘解得x e 解得12++=y y e x ,故)1ln(2++=x x y ,),(1+∞-∞=-f D（5）可解得⎩⎨⎧≤<--≤<-+=2122112/)1(y yy y x故⎪⎩⎪⎨⎧≤<--≤<-+=212211)1(21x x x x y ,]2,1(1-=-f D16. 某玩具厂每天生产60个玩具的成本为300元,每天生产80个玩具的成本为340元,求其线性成本函数,并求每天的固定成本和生产一个玩具的可变成本.解：设玩具的线性成本函数为bx a x C +=)(，则有⎩⎨⎧+=+=b a b a 8034060300 解得⎩⎨⎧==2180b a ，所以x x C 2180)(+= 故固定成本为180(元/每天),可变成本为2(元/每个).17. 某公司全年需购某商品2000台,每台购进价为5000元,分若干批进货.每批进货台数相同,一批商品售完后马上进下一批.每进货一次需消耗费用1000元,商品均匀投放市场（即平均年库存量为批量的一半）,该商品每年每台库存费为进货价格的%4.试将公司全年在该商品上的投资总额表示为批量的函数.解：设批量为x ，投资总额为y ，则x xy 1001021067+⨯+= 18. 某饲料厂日产量最多为m 吨,已知固定成本为a 元,每多生产1吨饲料,成本增加k 元.若每吨化肥的售价为p 元,试写出利润与产量x 的函数关系式.解：设利润为)(x L ，则a x k p x L --=)()( (元) ,],0[m x ∈19. 生产某种产品,固定成本为3万元,每多生产1百台,成本增加1万元,已知需求函数为p Q 210-=（其中p 表示产品的价格,Q 表示需求量）,假设产销平衡,试写出：（1）成本函数；（2）收入函数；（3）利润函数.解：(1) 3)(+=Q Q C (万元)(2) 2215)10(21)(Q Q Q Q P Q Q R -=⋅--=⋅= (万元) (3) 3421)()()(2-+--=Q Q Q C Q R Q L (万元) 20. 某酒店现有高级客房60套,目前租金每天每套200元则基本客满,若提高租金,预计每套租金每提高10元均有一套房间会空出来,试问租金定为多少时,酒店房租收入最大？收入多少元？这时酒店将空出多少套高级客房？解：设每套资金为x 元，酒店房租总收入为y 元，则有16000)400(101)1020060(2+--=--=x x x y ，故400=x 元/套,收入最大，为16000元, 这时酒店将空出20套高级客房.（B ）1. 设x x f x x f =-⎪⎭⎫⎝⎛-+)(212212,求)(x f . 解：令2212-+=x x t ，得2212-+=t t x ，有2212221221)(-+=⎪⎭⎫ ⎝⎛-+-t t t t f t f ,即2212221221)(-+=⎪⎭⎫ ⎝⎛-+-x x x x f x f 又()x x f x x f =--+21)2212(，可解得()11322-++=x x x x f 2. 设下面所考虑的函数都是定义在区间),(l l -上的,证明：（1）两个偶函数的和是偶函数,两个奇函数的和是奇函数；（2）两个偶函数的乘积是偶函数,两个奇函数的乘积是偶函数,偶函数与奇函数的乘积是奇函数.证明：设)(1x f 和)(2x f 为偶函数，)(1x g 和)(2x g 为奇函数， （1）设)()()(21x f x f x f +=)()()()()()(2121x f x f x f x f x f x f =+=-+-=-故)(x f 为偶函数，得证. 设)()()(21x g x g x g +=)()()()()()(2121x g x g x g x g x g x g -=--=-+-=-故)(x g 为奇函数，得证.（2）设)()()(21x f x f x h ⋅=)()()()()()(2121x h x f x f x f x f x h =⋅=-⋅-=-故)(x h 为偶函数，得证. 设)()()(21x g x g x I ⋅=[][])()()()()()(2121x I x g x g x g x g x I =-⋅-=-⋅-=-故)(x I 为偶函数，得证. 设)()()(11x g x f x J ⋅=[])()()()()()(1111x J x g x f x g x f x J -=-⋅=-⋅-=-故)(x J 为奇函数，得证.3. 设函数)(x f 和)(x g 在D 上单调增加,试证函数)()(x g x f +也在D 上单调增加.证明：设D x x ∈<21,[][][][]0)()()()()()()()(12121122>+-+=+-+x g x g x f x f x g x f x g x f∴函数)()(x g x f +也在D 上单调增加.4. 设函数)(x f 在区间],[b a 和],[c b 上单调增加,试证)(x f 在区间],[c a 上仍单调增加.证明: 设[]c a x x ,21∈<,若c x x ≤<21，由题意有)()(12x f x f >， 若21x x b <≤，由题意有)()(12x f x f >， 若21x b x <≤，则)()()(12x f b f x f ≥>，若21x b x ≤<，则)()()(12x f b f x f >≥， 综上，)(x f 在区间],[c a 上仍单调增加.5. 设函数)(x f 和)(x g 在D 上有界,试证函数)()(x g x f ±和)()(x g x f ⋅在D 上也有界.证明：由题)(x f 和)(x g 在D 上有界，即对D x ∈∀，0,021>>∃M M ，有1)(M x f ≤,2)(M x g ≤,则21)()(M M x g x f +≤+,21)()(M M x g x f ⋅≤⋅ 即函数)()(x g x f ±和)()(x g x f ⋅在D 上有界. 6. 证明函数x x y sin =在),0(+∞上无界.证明：对任意0>M ，都存在02[,]x M M π∈+使得1sin 0=x ，则M x x x >=000sin ，即函数x x y sin =在),0(+∞上无界.7. 设)(x f 为定义在),(l l -的奇函数,若)(x f 在),0(l 内单调增加,证明)(x f 在)0,(l -内也单调增加.证明：设)0,(21l x x -∈<，则),0(12l x x ∈-<-，)()()()()()(211212x f x f x f x f x f x f ---=-+--=-)(x f 在),0(l 内单调增加,∴0)()(12>-x f x f ，∴)(x f 在)0,(l -内也单调增加.8. 已知函数)(x f 满足如下方程：0,)1()(≠=+x xcx bf x af其中c b a ,,为常数,且b a ≠,求)(x f ,并讨论)(x f 的奇偶性.解：由已知，xc x bf x af =+)1()(， 令xt 1=，则有ct t bf t af =+)()1(，即cx x bf x af =+)()1(可解得)()(22xabx a b c x f --= ， 而)()(x f x f -=-,故)(x f 是奇函数.习题二答案(A)1. 观察判别下列数列的敛散性；若收敛,求其极限值：(1) nn u 31=; (2) 11ln +=n u n; (3) 212nu n +=; (4) 11+-=n n u n ;(5) nn u n πsin 1=; (6) n u n n )1(-=;(7) nn u )1(3-=; (8) πn nu n cos 1=. 解：(1) 收敛于0; (2) 发散; (3) 收敛于2; (4) 收敛于1; (5) 收敛于0; (6) 收敛于0; (7) 发散; (8) 收敛于0.2. 利用数列极限的分析定义证明下列极限： (1) 011lim=++∞→n n ; (2) 1311lim =⎪⎭⎫ ⎝⎛-+∞→n n ;(3) 532513lim =+++∞→n n n ; (4) 071lim =⎪⎭⎫⎝⎛-+∞→nn .（1）证明：0>∀ε，不妨设1<ε，要使ε<+=-110n u n 成立，只需112->εn 成立，因此取⎥⎦⎤⎢⎣⎡=21εN ，则当N n >时，有ε=<+=-N n u n 1110，所以011lim=++∞→n n .（2）证明：0>∀ε，要使ε<=-n u n 311成立，只需ε31>n 成立，因此取131+⎥⎦⎤⎢⎣⎡=εN ，则当N n >时，有ε<<=-N n u n 31311，即1)311(lim =-+∞→nn . （3）证明：0>∀ε，不妨设101<ε，取152251+⎥⎦⎤⎢⎣⎡-=εN ，则当N n >时，有ε<+<+=-)25(51)25(5153N n u n ，所以532513lim =+++∞→n n n .（4）证明：0>∀ε，不妨设1<ε，取11log 7+⎥⎦⎤⎢⎣⎡=εN ，则当N n >时，有ε<<=-N n n u 71710，所以071lim =⎪⎭⎫⎝⎛-+∞→nn .3. 求下列数列的极限：(1) 98124lim 22++-+∞→n n n n ; (2) 529lim 2+++∞→n n n n ; (3) nn n n n -+-++∞→32lim; (4) )5(lim 2n n n n -++∞→;(5) )11()311)(211(lim 222nn ---+∞→ ; (6) nnn 5151131311lim+++++++∞→ ; (7) )1sin (sin lim --+∞→n n n ; (8) nnn n n 1)4321(lim ++++∞→;(9) ⎪⎪⎭⎫ ⎝⎛+++⋅+⋅+∞→)1(1321211lim n n n ; (10) 11)1(6)1(6lim +++∞→-+-+n n nn n . （1）＝98124lim22++-+∞→n n n n 21/98/1/24lim 222=++-+∞→n n n n n （2）＝529lim2+++∞→n n n n 235219lim =+++∞→nn n （3）nn n n n -+-++∞→32lim32)2(3)3(2lim)2)(3)(3()3)(2)(2(lim =++++=++++-+++++-+=+∞→+∞→n n n n n n n n n n n n n n n n n n（4） )5(lim 2n n n n -++∞→2555lim 5)5)(5(lim2222=++=++++-++∞→+∞→n n n n n n n n n n n n n n n ＝ （5）因为 n n n n n n n 11)11)(11(112+⋅-=+-=-， 所以)11()311)(211(lim 222nn ---+∞→2121lim 11454334322321lim=+=+-⨯⨯⨯⨯⨯+∞→+∞→n n nn n n n n ＝（6）＝nnn 5151131311lim +++++++∞→ 565/1113/111lim =-÷-+∞→n（7）)1sin (sin lim --+∞→n n n21cos )1(21sin 2lim 21cos21sin2lim =-+-+=-+--+∞→+∞→n n n n n n n n n n ＝ （8）＝nnn n n 1)4321(lim ++++∞→4)43()42()41(1lim 41=⎥⎦⎤⎢⎣⎡++++∞→nn n n n （9） ⎪⎪⎭⎫⎝⎛+++⋅+⋅+∞→)1(1321211lim n n n 1)111(lim )111()3121()211(lim 4=+-=⎥⎦⎤⎢⎣⎡+-+-+-=+∞→+∞→n n n n n（10）=-+-++++∞→11)1(6)1(6lim n n n n n 61)6)1(1()6)1(61(lim 111=-+-+++++∞→n n n nn4. 判断下列结论是否正确,为什么？(1) 设数列}{n u ,当n 越来越大时,A u n -越来越小,则A u n n =+∞→lim ;(2) 设数列}{n u ,当n 越来越大时,A u n -越来越接近于零,则A u n n =+∞→lim ;(3) 设数列}{n u ,若对+∈∃>∀Z N ,0ε,当N n >时,有无穷多个n u 满足ε<-A u n ,则A u n n =+∞→lim ;(4) 设数列}{n u ,若对0>∀ε,}{n u 中仅有有限个n u 不满足ε<-A u n ,则A u n n =+∞→lim ;(5) 若}{n u 收敛,则k n n n n u u ++∞→+∞→=lim lim (k 为正整数);(6) 有界数列}{n u 必收敛； (7) 无界数列}{n u 必发散； (8) 发散数列}{n u 必无界.解： (1) 错; (2) 错; (3) 错; (4) 正确; (5)正确; (6) 错; (7) 正确; (8) 错.5. 利用函数极限的分析定义证明下列极限：(1) 539lim22=--→x x x ; (2) 0)21(lim =+∞→x x ; (3) 1)32(lim 2=-→x x ; (4) 02lim 2=-+→x x .证明：（1）0>∀ε，取εδ=，当δ<-<20x 时，有εδ=<-=--2392x x x ，故 539lim 22=--→x x x .（2）0>∀ε，不妨设1<ε，取ε1log 2=M ，则当M x >时，有ε=<M x )21()21(，故0)21(lim =+∞→x x .（3）0>∀ε，取2/εδ=，当δ<-<20x 时，有εδ=<-=--222132x x ，故 1)32(lim 2=-→x x .（4）0>∀ε，取2εδ=，当δ<-<20x 时，有εδ=<-=-22x x ，故 02lim 2=-+→x x .6. 下列函数什么过程中是无穷小量,什么过程中是无穷大量？(1) 21xy =; (2) )2ln()1(+-=x x y ; (3) xe y -=; (4) 2tan x y =;(5) xy -=112; (6) 12322-+-=x x x y . 解：(1) ∞→x 无穷小量,0→x 无穷大量;(2) 1→x 无穷小量,1-→x 无穷小量,+-→2x 无穷大量,+∞→x 无穷大量;(3) +∞→x 无穷小量 ,-∞→x 无穷大量;(4) πk x 2→(k 为整数)无穷小量 ,ππ+→k x 2(k 为整数)无穷大; (5) +→1x 无穷小量,-→1x 无穷大量; (6) 2→x 无穷小量,1-→x 无穷大量. 7. 求下列函数的极限：(1) 852)3)(sin 6(lim 32+--+∞→x x x x x x ; (2) 732523lim 42+--+∞→x x x x x ; (3) 12102)12()31(lim +-∞→x x x x ; (4) )2(lim 22++-∞→x x x x ; (5) 125lim 3++∞→x x x ; (6) 2)2sin(lim --∞→x x x ;(7) )1(lim 33x x x -+∞→; (8) xx x 1lim2++∞→; (9) xx x 1lim2+-∞→;(10) )49(lim +-++∞→x x x a a (0>a 且1≠a )解：（1）0852)3)(sin 6(lim 32＝+--+∞→x x x x x x （2）=+--+∞→732523lim 42x x x x x 23732523lim 432=+--+∞→xx x x x （3）=+-∞→12102)12()31(limx x x x 1210121012121210223)21()31(lim )12()31(lim =+-=+-∞→∞→x x x x x x x x x（4）)2(lim 22++-∞→x x x x122lim2)2)(2(lim 222222222-=+--=+-+-++=-∞→-∞→x x x xx x x x x x x x x x x（5）∞=++∞→125lim3x x x （6）02)2sin(lim=--∞→x x x（7）)1(lim 33x x x -+∞→)1(11lim)1(1))1(1)(1(lim32333232333232333233=-+-⋅-=-+-⋅--+-⋅--+=∞→∞→x x x x x x x x x x x x x x x x（8）11lim2=++∞→xx x （9）11lim2-=+-∞→xx x （10）当10<<a 时，=+-++∞→)49(lim x x x a a 149=-当1>a 时，)49(lim +-++∞→x x x a a049)49)(49(lim=+++++++-+=+∞→xxx x x x x a a a a a a8. 求下列函数的极限：(1) )153(lim 22--→x x x ; (2) 11lim 1--→n m x x x (n m ,为正整数);(3) 11lim31--→x x x ; (4) ⎪⎭⎫⎝⎛---→121lim 21x x xx ;(5) 22lim 2-→x xx ; (6) 3152lim 23--+→x x x x ;(7) 2211limx x x +-→; (8) ⎪⎭⎫⎝⎛+-++--→x x x x x x 212112lim ;(9) x x xx -----→111lim 1; (10) 1lim 21--+++→x nx x x n x .解：（1）1)153(lim 22＝--→x x x（2）＝11lim 1--→nm x x x nm x x x x x x n m x =+++-+++---→)1)(1()1)(1(lim 111（3）=--→11lim31x x x 32)1)(1)(1()1)(1)(1(lim33233231=+++-+++-→x x x x x x x x x （4）=⎪⎭⎫ ⎝⎛---→121lim 21x x xx 2312lim 12)1(lim 22121=--+=--+→→x x x x x x x x （5）由022lim2=-→xx x ，有∞=-→22lim 2x xx（6）=--+→3152lim 23x x x x 83)3)(5(lim 3=--+→x x x x（7）=+-→2211limx x x 2)1(1)11(lim 2220-=+-++→x x x x （8）=⎪⎭⎫⎝⎛+-++--→x x x x x x 212112lim 4)1(2lim 22lim 1221=-=+-+--→-→x x x x x x x x x （9）=-----→xx x x 111lim 11111lim 1-=---→x x（10）1lim 21--+++→x nx x x n x2)1(21)1()1(1[lim 1)1()1()1(lim 1121+=+++=+++++++=--++-+-=-→→n n n x x x x x x x n x n x9. 求下列各题中的常数a 和b :(1) 1112lim 23=⎪⎪⎭⎫ ⎝⎛++-+∞→x x b ax x ; (2) 51lim 21=-++→x abx x x ;(3) k b ax x x x =--+++∞→)1(lim 2(k 为已知常数).解：(1)因为11)2(11222323+-+++-=++-+x b ax bx x a x x b ax 若1)112(lim 23=++-+→∝x x b ax x ，则02=-a ，1=b ，即2=a ，1=b . （2）因为01lim )1(lim 1)1(lim )(lim 2112121=-++-=-++⋅-=++→→→→xa bx x x x a bx x x a bx x x x x x所以01=++b a ，即a b --=1511))(1(lim 1)1(lim 1lim 12121=-=---=-++-=-++→→→a xa x x x a x a x x a bx x x x x 故6=a ，7-=b . （3）因为kbax x x b x ab x a b ax x x x x =++++-+-+-=--++∝+→∝+→11)21()1(lim)1(lim 22222因此012=-a ，0>a ，k a ab =+-121，求得1=a ，k b -=21.10. 求下列函数极限： (1) x x x 3arcsin 4arctan lim0→; (2) xxx 3sin 2tan lim 0→;(3) x x x 1sin lim ∞→; (4) 2)4sin(lim 22--→x x x ;(5) 2220)cos 1(tan lim x x x x -→; (6) )1cos 1(lim 2xx x -∞→;(7) 30sin 1tan 1limxx x x +-+→; (8) x x xx x sin 3sin 2lim 0+-→;(9) x x x x 2sin 5tan lim0-→; (10) hxh x h sin )sin(lim 0-+→.解：（1）3434lim 3arcsin 4arctan lim 00＝＝x x x x x x →→（2）3232lim 3sin 2tan lim 00==→→x x x x x x（3）=∞→x x x 1sin lim 1/1)/1sin(lim =∞→xx x（4）=--→2)4sin(lim22x x x 424lim 22=--→x x x （5）=-→2220)cos 1(tan limx xx x 4)2/(lim 2240=→x x x （6）=-∞→)1cos1(lim 2x x x 2121lim 22=⋅∞→xx x（7）=+-+→30sin 1tan 1limx xx x 4121lim 212)cos 1(tan lim 32030=⋅=-→→xxx xx x x x （8）=+-→x x xx x sin 3sin 2lim041/)(sin 3/)(sin 2lim0=+-→x x x x x （9）=-→x xx x 2sin 5tan lim03252sin lim 5tan lim 00=-=-→→xx x x x x（10）=-+→h x h x h sin )sin(lim 0x hh x h h cos ]2/)2cos[()2/sin(2lim 0=+→11. 求下列函数极限： (1) xx x)11(lim -∞→; (2) x x x 2cot 20)(sec lim →;(3) 121011lim +→⎪⎭⎫⎝⎛+xx x ; (4) xx x x ⎪⎭⎫⎝⎛+-∞→22lim ;(5) 311lim +∞→⎪⎭⎫⎝⎛-+x x x x ; (6) xx x-→111lim .解：（1）=-∞→x x x )11(lim 1)1()11(lim ---∞→=-e xx x（2）=→xx x 2cot20)(sec lim e x xx =+→2tan120)tan 1(lim（3）121011lim +→⎪⎭⎫⎝⎛+xx x21)1(21100210)11(lim 11lim )11(lim -+⋅+→→→=+-=+⋅+=exxx x x x x x x x x（4）xx x x ⎪⎭⎫⎝⎛+-∞→22lim42)4(422)4(42)241(lim )241(lim )241(lim --∞→-⋅+-∞→--⋅+-∞→=+-⋅+-=+-=e x x x x x x x x （5）=⎪⎭⎫⎝⎛-++∞→311lim x x x x 212213)121(lim )11(lim )11(lim e x x x x x x x x x x =-+=-+⋅-++⋅-∞→∞→∞→（6）=-→xx x111lim 1)1(111)11(lim --⋅-→=-+e x x x12. 求下列函数极限： (1) )6sin(sin 21lim6ππ--→x x x ; (2) xxx 251ln lim0+→;(3) )21ln()31ln(lim x x x ++-∞→; (4) 1arcsin lim 20--→x x e xx ;(5) )1ln(121lim2x x x x ---→; (6) x e x x 21lim3sin 0-→;(7) xx x 1)tan 21(lim ++→; (8) xxx e x 10)(lim +→;(9) x x x x 3)421ln(lim 20+-→; (10) )4tan()2tan(lim 4x x x -⋅→ππ;(11) xx x 1)sin 1(lim -→; (12) x x x 2cot 10)(cos lim +→.解：（1）令6π-=x t ，6π→x 时0→t ，原式化为)6sin(sin 21lim6ππ--→x x x3)sin 3(lim cos 1lim ]2/)(cos 2/)(sin 3[21limsin )6/sin(21lim0000-=-+-=+-=+-→→→→tt t t tt t tt t t t t π＝（2）＝x x x 251ln lim0+→45252122/)51ln(lim 0=⋅=+→x x x（3）＝)21ln()31ln(limx x x ++-∞→0)23(lim 23lim ==-∞→-∞→xx x x x （4）＝1arcsin lim2--→xx e x x 1lim220-=-→x x x （5）=---→)1ln(121lim 20x x x x 12/)2(lim 220=--→xx x（6）＝xe x x 21lim3sin 0-→6123/)(sin lim 0-=-→x x x（7）xx x 1)tan 21(lim ++→ 2tan 2limtan 210tan 22tan 2100)tan 21(lim )tan 21(lim e x x xxx x x x x x x =+=+=+→++⋅→⋅→（8）=+→xxx e x 1)(lim 2111)11(lim e e x x e x e x x x x x =-++-+⋅-+→（9）=+-→x x x x 3)421ln(lim2032324lim 20-=-→x x x x （10）令x t -=4π，4π→x 时0→t ，)4tan()2tan(lim 4x x x -⋅→ππ21tan 2tan 1lim 2cot lim )22/tan(lim 2000=-⋅==⋅-=→→→t t t t t tt t t t π（11）=-→xx x 10)sin 1(lim 1sin sin 10)sin 1(lim --⋅-→=-e x xx x x（12）xx x 2cot 10)(cos lim +→21tan 1cos 1cos 10cot 022)1cos 1(lim cos lim cos lim --⋅-→→→=-+=⋅=ex xx xx x x xx x13. 证明：0)2(1)1(11lim 222=⎥⎦⎤⎢⎣⎡+++++∞→n n n n . 证明：222221)2(1)1(11)2(1n n n n n n n +≤++++≤+ 又由01lim )2(1lim22=+=++∞→+∞→n n n n n n所以0])2(1)1(11[lim 222=+++++∞→n n n n 14．求下列函数的间断点,并判断类型.)1( 1212)(11+-=xxx f ; (2) ()x x x x f 21)1ln()(2--=;(3) ⎪⎩⎪⎨⎧-=-≠+-=10111)(2x x xx x f ; +++++++++++++++++++++++ (4) ⎪⎩⎪⎨⎧≥+<≤+<=23212416)(2x x x x x x f 解：（1）11212lim/1/10-=+--→x x x 12/112/11lim 1212lim /1/10/1/10=+-=+-++→→xxx x x x即0=x 为跳跃间断点.（2）0)21()1ln(lim20=--→x x x x ，即0=x 为可去间断点. 0)21()1ln(lim 20=--→x x x x ，即21=x 为无穷间断点. （3）211lim 21=+--→xx x ，即1-=x 为可去间断点.（4）10)(lim 2=-→x f x ，7)(lim 2=+→x f x ，即2=x 为跳跃间断点. 15. 讨论下列函数的连续性:(1) ⎪⎩⎪⎨⎧≥<=)0(0)0(1sin)(2x x xx x f ;(2) ⎪⎩⎪⎨⎧=≠=)0(1)0(sin )(x x xx x f ;(3) ⎪⎩⎪⎨⎧=≠=003)(1x x x f x. 解：(1) 0<x 时，xx x f 1sin)(2=连续， 0>x 时，0)(=x f 连续，0=x 时，)(lim 0)(lim 0_x f x f x x +→→==连续， 所以)(x f 在),(+∞-∞连续.（2） 0<x 时，x xx f sin )(-=连续， 0>x 时，xxx f sin )(=连续，0=x 时，1)sin (lim )(lim 00-=-=--→→xxx f x x ，1sin lim )(lim 00==++→→xxx f x x ， 所以)(x f 在0=x 处不连续.（3）0≠x 时，xx f 13)(=连续，03lim )(lim 10==--→→x x x x f ，∞==++→→xx x x f 103lim )(lim ， 所以)(x f 在0=x 处不连续. 16. 确定常数b a ,使下列函数连续：(1) ⎪⎩⎪⎨⎧+<<--=其他53541)(2bxa x x x f ; (2) ⎪⎪⎩⎪⎪⎨⎧>+=-<=01sin 010sin 1)(x b x x x a x xx x f .解：(1)若)(x f 在54-=x ，53=x 处连续，则有2)54()54(1lim)(lim x bx a x x -=++--→-→，)(lim 1lim)53(2)53(bx a x x x +=-+-→→，即⎪⎪⎩⎪⎪⎨⎧=+=-54535354b a b a ，解得⎪⎪⎩⎪⎪⎨⎧==7175b a （2）)(x f 在0=x 处连续，1sin 1lim 0-=-→a x x x ，1)sin 1(lim 0-=++→a b x xx ， 有11=-a ，1-=a b ，解得2=a ，1=b . 17. 试证方程0133=--x x 在区间)2,1(内至少有一个实根. 证明：令13)(3--=x x x f ，)(x f 在)2,1(连续，03)1(<-=f ，01)2(>=f ，由零点定理知，至少存在一点)2,1(0∈x ，使得0)(0=x f 成立，即 方程0133=--x x 在区间)2,1(内至少有一个实根. 18. 试证方程2-=xe x 在区间)2,0(内至少有一个实根. 证明：令2)(+-=xe x xf ，)(x f 在)2,0(连续，01)0(>=f ，04)2(2<-=e f ，由零点定理知，至少存在一点)2,0(0∈x ，使得0)(0=x f 成立，即 方程2-=xe x 在区间)2,0(内至少有一个实根.(B)1. 求极限)31ln()21ln(lim x x x +++∞→.解：)31ln()21ln(lim x x x +++∞→3ln 2ln )3/11ln(3ln )2/11ln(2ln lim)3/11ln(3ln )2/11ln(2ln lim=++++=+++++∞→+∞→x xx x x x x x x x ＝2. 设nn n n n u n ++++++=2222211 ,求n n u +∞→lim . 解：因为1212122++++≤≤++++n n u n n n n ， 212/)1(lim 21lim 22=++=++++∝+→∝+→nn n n n n n n n ， 2112/)1(lim 121lim 22=++=++++∝+→∝+→n n n n n n n ， 由极限存在定理可知，21lim =∝+→n n u . 3. 设数列}{n u ：,2,,222,22,21-++++n u ,证明：n n u +∞→lim 存在,并求此极限值.证明：首先证}{n u 单调增加。
大一经济数学习题答案大一经济数学习题答案在大一学习经济数学时，习题是我们巩固知识、提高能力的重要方式之一。
然而，有时候我们可能会遇到一些难题，无法得到正确的答案。
本文将为大一经济数学中一些常见的习题提供答案，并对解题思路进行简要的分析。
一、微积分1. 计算函数 f(x) = 2x^3 - 5x^2 + 3x - 7 的导数。
解答：对于多项式函数 f(x) = 2x^3 - 5x^2 + 3x - 7，我们可以按照幂次逐项求导。
首先，对于 x^n，其导数为 n*x^(n-1)。
因此，我们可以得到 f'(x) = 6x^2 - 10x + 3。
2. 求函数 f(x) = e^x * ln(x) 的导数。
解答：根据指数函数和对数函数的导数公式，我们可以得到 f'(x) = e^x * ln(x) + e^x / x。
3. 求函数f(x) = ∫(0,x) t^2 dt 的导数。
解答：根据牛顿-莱布尼茨公式，我们可以将该函数视为一个定积分的上限函数，即 f(x) = F(x) - F(0)，其中F(x) = ∫(0,x) t^2 dt。
根据定积分的基本性质，我们可以得到 f'(x) = F'(x) - F'(0) = x^2 - 0 = x^2。
二、线性代数1. 求矩阵 A = [1 2; 3 4] 的逆矩阵。
解答：我们可以使用矩阵的伴随矩阵求逆矩阵。
首先，计算矩阵 A 的行列式为|A| = 1*4 - 2*3 = -2。
然后，计算矩阵 A 的伴随矩阵为 A* = [4 -2; -3 1]。
最后，根据逆矩阵的定义，我们可以得到 A 的逆矩阵为 A^-1 = A*/|A| = [4/-2 -2/-2;-3/-2 1/-2] = [-2 1; 3/2 -1/2]。
2. 求向量 v = [1; 2; 3] 在向量空间 span{[1; 0; 0], [0; 1; 0]} 中的投影向量。
