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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.。
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所以，r(/)A有固运方向.6.向绘歯数半行于固立屮面的充摆杀件是(F尹产)司卩分析:向呈诵数？W平If于固定平面的充要余件是存在•牛定向向蚩50*使？(心 = 0 ,所以我们蹩耳求这个向旅亓及万与尹.严的尢系"证若尺刀半苻于個址羊面—设乔足¥面斗的•个单位迖向嵐则习为常向議H?(/) 7t-0 -两次求微商色尸7 =0・?y 7i=0 ,即问最孑，戸‘唾直于同•非零向輦无因而典而*即(F戶尹')刃.反之，若(? r1 F M) =0i则有r x ?=6戒产x戸工6 .若产x? = 0i由匕题柯产(/) 具冇■的崔方向、白然半fr于一固宦半面，若rx? H 0(则存圧数母焰数入(“、H&n使戸'= 乔*尹①令聞*厂桁丰6,且;V)丄讯/)* 4^7 X?求微商井将①式代入得用=Fx P*—/I t r X r1)—p f是x ^' —6 .市上题划另4fhM眾方向，而F(f)丄苑即巩f) 平存于固進半而S3曲线的概念1-求圆柱螺^T=cosr- ,F=sinr, f *在(1Q 0)的切线和注平面。
第一章 函数极限与连续一、填空题1、已知x x f cos 1)2(sin +=，则=)(cos x f 。
2、=-+→∞)1()34(lim22x x x x 。
3、0→x 时，x x sin tan -是x 的阶无穷小。
4、01sin lim 0=→xx k x 成立的k 为。
5、=-∞→x e x x arctan lim 。
6、⎩⎨⎧≤+>+=0,0,1)(x b x x e x f x 在0=x 处连续，则=b 。
7、=+→xx x 6)13ln(lim 0。
8、设)(x f 的定义域是]1,0[，则)(ln x f 的定义域是__________。
9、函数)2ln(1++=x y 的反函数为_________。
10、设a 是非零常数，则________)(lim =-+∞→xx ax a x 。
11、已知当0→x 时，1)1(312-+ax 与1cos -x 是等价无穷小，则常数________=a 。
12、函数xxx f +=13arcsin)(的定义域是__________。
13、lim ____________x →+∞=。
14、设8)2(lim =-+∞→xx ax a x ，则=a ________。
15、)2)(1(lim n n n n n -++++∞→=____________。
二、选择题1、设)(),(x g x f 是],[l l -上的偶函数，)(x h 是],[l l -上的奇函数，则中所给的函数必为奇函数。
（Ａ）)()(x g x f +；（Ｂ）)()(x h x f +；（C ）)]()()[(x h x g x f +；（D ）)()()(x h x g x f 。
2、xx x +-=11)(α，31)(x x -=β，则当1→x 时有。
（Ａ）α是比β高阶的无穷小； （Ｂ）α是比β低阶的无穷小；（C ）α与β是同阶无穷小； （D ）βα~。
3、函数⎪⎩⎪⎨⎧=-≥≠-+-+=0)1(0,1111)(3x k x x x x x f 在0=x 处连续，则=k 。
微积分各章习题及详细答案(供参考)第一章函数极限与连续一、填空题1、已知 f (sin x) 1cos x ，则 f (cos x)。
2(4 3x)22、 lim2)。
xx(1 x3、 x 0 时， tan x sin x 是 x 的阶无量小。
4、 lim xksin10 建立的 k 为。
xx5、 lim e x arctan xx6、 f ( x)ex1, xb,7、 limln( 3x1)x 06x。
x 0在 x 0处连续，则 b 。
x 0。
8、设 f (x) 的定义域是 [ 0,1] ，则 f (ln x) 的定义域是 __________ 。
9、函数 y 1 ln( x 2) 的反函数为 _________。
10、设 a 是非零常数，则 lim (xa) x ________ 。
xx a111、已知当 x 0时， (1 ax 2 ) 3 1与 cosx 1 是等价无量小，则常数 a ________。
12、函数 f ( x)arcsin3x的定义域是 __________ 。
1 x13、 lim ( x 22x 2 2)____________ 。
x14、设 lim (x2a ) x 8 ，则 a________。
xx a15、 lim ( n n 1)( n 2n) =____________ 。
n二、选择题1、设 f ( x), g(x) 是 [ l , l ] 上的偶函数， h( x) 是 [ l , l ] 上的奇函数，则中所给的函数必为奇函数。
（Ａ） f ( x) g( x) ；（Ｂ） f ( x) h( x) ；（ C ） f (x)[ g(x) h( x)] ；（ D ） f ( x) g( x) h(x) 。
2、1 x3x( x)，( x)1x ，则当时有。
1 x1（Ａ） 是比 高阶的无量小； （Ｂ） 是比 低阶的无量小；（ C ）与 是同阶无量小；（ D ）~。
3、函数 f (x)1 x 1 ,x 0( x1) 在 x0处连续，则 k3 1 x 1 。
第一章 函数极限与连续一、填空题1、已知x x f cos 1)2(sin +=，则=)(cos x f 。
2、=-+→∞)1()34(lim22x x x x 。
3、0→x 时，x x sin tan -是x 的阶无穷小。
4、01sin lim 0=→xx k x 成立的k 为。
5、=-∞→x e x x arctan lim 。
6、⎩⎨⎧≤+>+=0,0,1)(x b x x e x f x 在0=x 处连续，则=b 。
7、=+→xx x 6)13ln(lim 0。
8、设)(x f 的定义域是]1,0[，则)(ln x f 的定义域是__________。
9、函数)2ln(1++=x y 的反函数为_________。
10、设a 是非零常数，则________)(lim =-+∞→xx ax a x 。
11、已知当0→x 时，1)1(312-+ax 与1cos -x 是等价无穷小，则常数________=a 。
12、函数xxx f +=13arcsin)(的定义域是__________。
13、lim ____________x →+∞=。
14、设8)2(lim =-+∞→x x ax a x ，则=a ________。
15、)2)(1(lim n n n n n -++++∞→=____________。
二、选择题1、设)(),(x g x f 是],[l l -上的偶函数，)(x h 是],[l l -上的奇函数，则中所给的函数必为奇函数。
（Ａ）)()(x g x f +；（Ｂ）)()(x h x f +；（C ）)]()()[(x h x g x f +；（D ）)()()(x h x g x f 。
2、xx x +-=11)(α，31)(x x -=β，则当1→x 时有。
（Ａ）α是比β高阶的无穷小； （Ｂ）α是比β低阶的无穷小；（C ）α与β是同阶无穷小； （D ）βα~。
《微观经济学》〔高鸿业第四版〕答案第二章1. 已知某一时期内某商品的需求函数为Q d=50-5P，供应函数为Q s=-10+5p。
（1）求均衡价格P e和均衡数量Q e，并作出几何图形。
（2）假定供应函数不变，由于消费者收入水平提高，使需求函数变为Q d=60-5P。
求出相应的均衡价格P e和均衡数量Q e，并作出几何图形。
（3）假定需求函数不变，由于生产技术水平提高，使供应函数变为Q s=-5+5p。
求出相应的均衡价格P e和均衡数量Q e，并作出几何图形。
（4）利用〔1〕〔2〕〔3〕，说明静态分析和比较静态分析的联系和区别。
（5）利用〔1〕〔2〕〔3〕，说明需求变动和供应变动对均衡价格和均Q d衡数量的影响.解答:(1)将需求函数d Q= 50-5P和供应函数s Q=-10+5P代入均衡条件d Q= s Q,有:50- 5P= -10+5P得: P e=6以均衡价格P e =6代入需求函数d Q=50-5p ,得:Q e=50-520⨯6=或者,以均衡价格Pe =6 代入供应函数s Q=-10+5P ,得:Qe=-10+5206=⨯所以,均衡价格和均衡数量分别为Pe =6 , Qe=20 ...如图1-1所示. (2) 将由于消费者收入提高而产生的需求函数d Q =60-5p 和原供应函数s Q =-10+5P, 代入均衡条件d Q =s Q ,有: 60-5P=-10=5P得7=Pe以均衡价格 7=Pe 代入dQ =60-5p ,得Qe=60-5257=⨯或者,以均衡价格7=Pe 代入sQ =-10+5P, 得Qe=-10+5257=⨯所以,均衡价格和均衡数量分别为7=e P ，25=Qe (3) 将原需求函数d Q =50-5p 和由于技术水平提高而产生的 供应函数Q s =-5+5p ,代入均衡条件d Q =s Q ,有: 50-5P=-5+5P 得 5.5=e P以均衡价格5.5=e P 代入d Q =50-5p ,得5.225.5550=⨯-=e Q或者,以均衡价格5.5=e P 代入s Q =-5+5P ,得5.225.555=⨯+-=e Q所以,均衡价格和均衡数量分别为5.5=e P ，5.22=Qe .如图1-3所示. (4)所谓静态分析是考察在既定条件下某一经济事物在经济变量的相互作用下所实现的均衡状态及其特征.也可以说,静态分析是在一个经济模型中根据所给的外生变量来求内生变量的一种分析方法.以(1)为Pe-例,在图1-1中,均衡点E 就是一个表达了静态分析特征的点.它是在给定的供求力量的相互作用下所到达的一个均衡点.在此,给定的供求力量分别用给定的供应函数 s Q =-10+5P 和需求函数d Q =50-5p 表示,均衡点E 具有的特征是:均衡价格6=e P 且当6=e P 时,有d Q =s Q =20=Qe ;同时,均衡数量 20=Qe ,切当20=Qe 时,有e s d P P P ==.也可以这样来理解静态分析:在外生变量包括需求函数的参数(50,-5)以及供应函数中的参数(-10,5)给定的条件下,求出的内生变量分别为6=e P ，20=Qe 依此类推,以上所描素的关于静态分析的基本要点,在(2)及其图1-2和(3)及其图1-3中的每一个单独的均衡点()2,1i E 都得到了表达. 而所谓的比较静态分析是考察当所有的条件发生变化时,原有的均衡状态会发生什么变化,并分析比较新旧均衡状态.也可以说,比较静态分析是考察在一个经济模型中外生变量变化时对内生变量的影响,并分析比较由不同数值的外生变量所决定的内生变量的不同数值,以(2)为例加以说明.在图1-2中,由均衡点 变动到均衡点 ,就是一种比较静态分析.它表示当需求增加即需求函数发生变化时对均衡点的影响.很清楚,比较新.旧两个均衡点 和 可以看到:由于需求增加由20增加为25.也可以这样理解比较静态分析:在供应函数保持不变的前提下,由于需求函数中的外生变量发生变化,即其中一个参数值由50增加为60,从而使得内生变量的数值发生变化,其结果为,均衡价格由原来的6上升为7,同时,均衡数量由原来的20增加为25.类似的,利用(3)及其图1-3也可以说明比较静态分析方法的基本要求. 〔5〕由(1)和(2)可见,当消费者收入水平提高导致需求增加,即表现为需求曲线右移时,均衡价格提高了,均衡数量增加了.由(1)和(3)可见,当技术水平提高导致供应增加,即表现为供应曲线右移时,均衡价格下降了,均衡数量增加了.总之,一般地有,需求与均衡价格成同方向变动,与均衡数量成同方向变动;供应与均衡价格成反方向变动,与均衡数量同方向变动.2. 假定表2—5是需求函数Qd=500-100P 在一定价格范围内的需求表：某商品的需求表〔1〕求出价格2元和4元之间的需求的价格弧弹性。
微分几何第四版答案第一部分曲线与曲面的局部微分几何第一章欧氏空间1.1 向量空间1.2 欧氏空间第二章曲线的局部理论2.1 曲线的概念2.2 平面曲线2.3 E的曲线2.4 曲线论基本定理第三章曲面的局部理论3.1 曲面的概念3.2 曲面的第一基本形式3.3 曲面的第二基本形式3.4 法曲率与weingarten变换3.5 主曲率与Gauss曲率3.6 曲面的一些例子第四章标架与曲面论基本定理4.1 活动标架4.2 自然标架的运动方程4.3 曲面的结构方程4.4 曲面的存在惟一性定理4.5 正交活动标架4.6 曲面的结构方程（外微分法）第五章曲面的内蕴几何学5.1 曲面的等距变换5.2 曲面的协变微分5.3 测地曲率与测地线5.4 测地坐标系5.5 Gauss-Bonnet公式5.6 曲面的Laplace算子5.7 Riemann度量第二部分整体微分几何选讲第六章平面曲线的整体性质6.1 平面的闭曲线6.2 平面的凸曲线第七章曲面的若干整体性质7.1 曲面的整体描述7.2 整体的Gauss-Bonnet公式7.3 紧致曲面的Gauss映射7.4 凸曲面7.5 曲面的完备性第八章常Gauss曲率曲面8.1 常正Gauss曲率曲面8.2 常负Gauss曲率曲面与sine-Gordon方程8.3 Hilbert定理8.4 Backlund变换第九章常平均曲率曲面9.1 Hopf微分与Hopf定理9.2 Alexsandrov惟一性定理9.3 附录：常平均曲率环面第十章极小曲面10.1 极小图10.2 极小曲面的weierstrass表示10.3 极小曲面的Gauss映射10.4 面积的变分与稳定极小曲面索引。
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。
《微观经济学》（高鸿业第四版）答案第二章1.已知某一时期内某商品的需求函数为Q d=50-5P，供给函数为Q s=-10+5p。
（1）求均衡价格P e和均衡数量Q e，并作出几何图形。
（2）假定供给函数不变，由于消费者收入水平提高，使需求函数变为Q d=60-5P。
求出相应的均衡价格P e和均衡数量Q e，并作出几何图形。
（3）假定需求函数不变，由于生产技术水平提高，使供给函数变为Q s=-5+5p。
求出相应的均衡价格P e和均衡数量Q e，并作出几何图形。
（4）利用（1）（2）（3），说明静态分析和比较静态分析的联系和区别。
（5）利用（1）（2）（3），说明需求变动和供给变动对均衡价格和均Q d衡数量的影响.解答:(1)将需求函数d Q = 50-5P和供给函数s Q =-10+5P代入均衡条件d Q = s Q,有:50- 5P= -10+5P得: P e=6以均衡价格P e =6代入需求函数d Q=50-5p ,得:Q e=50-5206=⨯或者,以均衡价格 Pe =6 代入供给函数s Q =-10+5P ,得:Qe=-10+5206=⨯所以,均衡价格和均衡数量分别为Pe =6 , Qe=20 ...如图1-1所示. (2) 将由于消费者收入提高而产生的需求函数d Q =60-5p 和原供给函数s Q =-10+5P, 代入均衡条件d Q =s Q ,有: 60-5P=-10=5P得7=Pe以均衡价格 7=Pe 代入dQ =60-5p ,得Qe=60-5257=⨯或者,以均衡价格7=Pe 代入sQ =-10+5P, 得Qe=-10+5257=⨯所以,均衡价格和均衡数量分别为7=e P ，25=Qe(3) 将原需求函数d Q =50-5p 和由于技术水平提高而产生的 供给函数Q s =-5+5p ,代入均衡条件d Q =s Q ,有: 50-5P=-5+5P 得 5.5=e P以均衡价格5.5=e P 代入d Q =50-5p ,得5.225.5550=⨯-=e Q或者,以均衡价格5.5=e P 代入s Q =-5+5P ,得5.225.555=⨯+-=e Q所以,均衡价格和均衡数量分别为5.5=e P ，5.22=Qe .如图1-3所示. (4)所谓静态分析是考察在既定条件下某一经济事物在经济变量的相互作用下所实现的均衡状态及其特征.也可以说,静态分析是在一个经济模型中根据所给的外生变量来求内生变量的一种分析方法.以(1)Pe-为例,在图1-1中,均衡点E 就是一个体现了静态分析特征的点.它是在给定的供求力量的相互作用下所达到的一个均衡点.在此,给定的供求力量分别用给定的供给函数 s Q =-10+5P 和需求函数d Q =50-5p 表示,均衡点E 具有的特征是:均衡价格6=e P 且当6=e P 时,有d Q =s Q =20=Qe ;同时,均衡数量 20=Qe ,切当20=Qe 时,有e s d P P P ==.也可以这样来理解静态分析:在外生变量包括需求函数的参数(50,-5)以及供给函数中的参数(-10,5)给定的条件下,求出的内生变量分别为6=e P ，20=Qe 依此类推,以上所描素的关于静态分析的基本要点,在(2)及其图1-2和(3)及其图1-3中的每一个单独的均衡点()2,1i E 都得到了体现.而所谓的比较静态分析是考察当所有的条件发生变化时,原有的均衡状态会发生什么变化,并分析比较新旧均衡状态.也可以说,比较静态分析是考察在一个经济模型中外生变量变化时对内生变量的影响,并分析比较由不同数值的外生变量所决定的内生变量的不同数值,以(2)为例加以说明.在图1-2中,由均衡点 变动到均衡点 ,就是一种比较静态分析.它表示当需求增加即需求函数发生变化时对均衡点的影响.很清楚,比较新.旧两个均衡点 和 可以看到:由于需求增加由20增加为25.也可以这样理解比较静态分析:在供给函数保持不变的前提下,由于需求函数中的外生变量发生变化,即其中一个参数值由50增加为60,从而使得内生变量的数值发生变化,其结果为,均衡价格由原来的6上升为7,同时,均衡数量由原来的20增加为25.类似的,利用(3)及其图1-3也可以说明比较静态分析方法的基本要求.（5）由(1)和(2)可见,当消费者收入水平提高导致需求增加,即表现为需求曲线右移时,均衡价格提高了,均衡数量增加了.由(1)和(3)可见,当技术水平提高导致供给增加,即表现为供给曲线右移时,均衡价格下降了,均衡数量增加了.总之,一般地有,需求与均衡价格成同方向变动,与均衡数量成同方向变动;供给与均衡价格成反方向变动,与均衡数量同方向变动.2. 假定表2—5是需求函数Qd=500-100P 在一定价格范围内的需求表：某商品的需求表（1）求出价格2元和4元之间的需求的价格弧弹性。
参考答案与提示习题1-21、7)0(=f ；27)4(=f ；9)21(=-f ；732)(2+-=a a a f ；62)1(2++=+x x x f2、1)2(-=-f ；0)1(=-f ；1)0(=f ；2)1(=f3、（1）[)(]1,00,1 -；（2）1>x （3）[]3,1- （4）()()()+∞∞-,22,11,4、（1）x y 2cos 2+=（2）23cot x arc y =习题1-31. （1）5；（2）1；（3）不存在；（4）不存在 2．（1）2；（2）25；（3）23；（4）32-；（5）12-；（6）1. 习题1-41. (1)无穷小；（2）无穷大；（3）无穷大（∞-）；（4）-→0x 时是无穷小；+→0x 时是无穷大；2. （1）同阶无穷小；（2）高阶无穷小；（3）等价无穷小3. （1）1；（2）21；（3）23；（4）1 习题1-5（1）．24;（ 2）.0;（ 3）.35；（4）.∞；（5）.503030532⋅；（6）.21-；（7）.0；（8）.1259-；（9）.24925+；（10）.0 习题1-61．（1）35；（2）1x xsin lim x -=-→ππ；（3）4；（4）32（5）2；（6）2 2．（1）8e ；（2）1-e ；（3）32-e；（4）2-e （5）5e ；（6）e习题1-71.1=a ；1=b2.（1）1±=x 是第二类间断点中无穷间断点；（2）0x =是第二类间断点中的无穷间断点；（3）1=x 是第一类间断点中可去间断点；（4）1-=x 是第二类间断点中的无穷间断点，1=x 是第一类间断点中的跳跃间断点3.（1）)1ln(+e ；（2）232；（3）e a log 3；（4）1 复习题一1、（1）1；（2）[]2,1)0,2(⋃-；（3）[)3,0；（4）3；（5）ke ；（6）23；（7）2；（8）第一类间断点且可去间断点2、（1）C ；（2C （A.1x y -=；1x y .C --=）；（3）B ；（4）B ；（5）C ；（6）D ；（7）A ；（8）A3、（1）34；（2）312x x )1x sin(21x lim =-+-→；（3）2-e ；（4）1)x (sin x sin 330x lim =→；（5）31；（6）0)2x (sin xx 3x 2x lim=+-+∞→；（7）a cos ；（8）4π-4、1=a5、23=a 6、6b ,4a == 7、（1）21；（2）a 28、（1）11=x 是第一类间断点且是可去间断点，22=x 是第二类型无穷间断点；（2）01=x 是第一类间断点且是可去间断点，)(22Z k k x ∈+=ππ是第二类型无穷间断点；（3）0=x 是第一类间断点且是可去间断点；（4）0=x 是第一类间断点且是跳跃间断点 9、1=a习题2-11、(1) √ (2) × (3) × (4) × (5) × (6)、√2、2126()v t t =+∆+∆ 0.10.012|12.61|12.0601|12t t t v v v ∆=∆=====3、()2f x '=4、 (1) 在0x =处连续且可导(2) 在0x =处连续，但不可导5、切线方程：210x y --= 法线方程：230x y +-=6、t t d dtθ=7、dT dt习题2-21、 (1) × (2) × (3)、× (4)、√ (5)、×2、 (1) (0)0()2f f ππ''== (2) (0)1()1f f π''==- (3) (0)0(1)13f f ''== (4) 11(1)(4)418f f ''=-=-3、略4、 (1)2664x x ++ (2)212ln 2xx -(3)12632220xx x -----(4)1cos x x +(5)(ln sin cos )xa a x x ⋅+ (6)1cos ln sin x x x x⋅+(7)2983x x +- (8) 22(2tan 2sec )sec x x x x x ++(9) 31221122x x ---- (10)2sin 1cos x x x x ++-(11) 11222(1)x xx -+-- (12)22cos (sin 1)x x -- (13) cos 1sin x x x -+ (14) 22sin cos cos (1)x x x x x x +++(15)122ln 22xxx x --- (16)3cos 2sin 2x x xx- 5、切线方程：ln 210x y -+= 法线方程：ln 2ln 20x y +-= 6、切点坐标：(1,1)-- 切线方程：20x y ++= 法线方程：0x y -=习题2-31、(1)√ (2) × (3)× (4) ×2、(1) 2(41)xe x x ++(2) (3) tan x -(4) 23ln (1)+1x x + (5))1x ln n (nx 1n +- (6) 222sin 2sin 2sin cos x x x x x +(7)(8) (9) 24()x x e e ---(10)arcsin x(11)(12) 2242(1)16x x x -++ 3、()(1)(4)824f x f f '''===4、切线方程：20x y e --= 法线方程：230x y e +-= 5、30x y --习题2-41、(1)223(1)a y - (2)x ayax y+-+ (3)x y x y e y e x ---+ (4)21y xy - (5)y y e x -+ (6)cos()cos()x y x y e y xy e x xy +++-+2、 (1)232(2)31y y y x x x +-+-+ (2)cot 224(1)xxy y ye x x e +-- (3)(cos ln cos sin tan )y x x x x - (4) ln(5)5xyy x x -+-+ 3、(1)232te - (2) tan t 4、32t dydx π==-- 5、 (1)在0x =处切线方程：210x y +-= 法线方程：220x y -+=(2)在2t =处切线方程：43120x y a +-= 法线方程：3460x y a -+=习题2-51、 (1) 221(ln 3)3xx -(2) 22csc cot x x ⋅ (3)22(arctan )1x x x ++ (4) 2sec (tan sec )x x x + (5) -322(1)x x -+ (6) 21(ln 1)x x x x x-++2、(1) (1)7,(1)4,(1)0f f f ''''''=== (2)11(1),(0)2,(1)22f f f ''''''-==-= 3、 (1)0 (2) 3(ln3)xn(3)()11(2)!ln 1(1)(3)n n n n y x y y n xx--'''=+==-⋅≥ (4) ()xn x e + (5) 12cos(2)2n y x n π-=+⋅(6) 11(1)!5n ny n x +⎛⎫=- ⎪+⎝⎭4、略5、 (1)(4)4sin x ye x =-(2) (5)22sin cos 16cos y x x x x x =-- (3)(20)0y = 6、31cot 3,sin 3a θθ--。
