PART 3
TABLE OF CONTENTS
Introduction page 2 Sample questions page 3- 21 Answers to sample questions page 22-37 Content specification outline page 38-40 IMA Ethical Standards page 41-43
Attachments:
Time Value of Money Tables page 44-47 Order forms for additional resources page 48-50
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Introduction
The Certified Management Accountant (CMA) is endorsed by the Institute of Management Accountants (IMA).
The Institute of Certified Management Accountants (ICMA) is publishing this book of sample questions with answers to help you prepare for the CMA examination. This book is intended to familiarize you with the types of questions that you will encounter on the exams. These questions are actual “retired” questions from the computer-based CMA exam. This publication should be used as a supplement to other study materials.
The book includes 55 sample questions, with a proportional representation from each of the sections within this part and at various cognitive levels. Recorded next to each question number is a topic reference to the content specification outline, which is also included in this publication. In addition, we have enclosed time value of money tables that are available to you on the computer when you take the exam. The answers to the questions, along with explanations and supporting calculations, are shown at the end of the question section. Textbook references and the relevant learning outcome statements are also included with the answer.
The ICMA publishes a book of sample questions for each of the three multiple-choice parts of the revised CMA programs. We also publish “Resource Guide for Revised CMA Exam,” which includes comprehensive textbook references for all line items on the content specification outline, exam statistics, and exam taking strategies.
In addition, those candidates who previously took and failed an exam part can order an Advanced Exam Performance report. This report follows the Examination Content Specification Outlines, which means you will know exactly what topics and subtopics you need to study. The report separately shows areas of study that need improvement based on your exam results, so that you can save time by focusing on your weakest areas and improve your score.
The CMA Program is a rigorous test of your skills and capabilities and requires dedication to be successful. We hope that these sample questions will be a valuable resource as your pursue your goal of certification. Should you have questions about the certification process, please let us know or visit our website https://www.doczj.com/doc/bd14492252.html,. Good luck on the exam!
SAMPLE QUESTIONS – PART 3
1. Topic A.1.d.
Vasil Inc. conducted a strategic self-assessment of factors contributing to market attractiveness and business strengths as follows.
Market attractiveness Weight Rating (1-5)
Overall market size 0.4 5
Annual market growth rate 0.3 4
Historical profit margin 0.2 5
Business strength Weight Rating (1-5)
Unit costs 0.5 4
Market share growth 0.4 2
Brand reputation 0.1 3
The factor ratings range from 1 (the lowest) to 5 (the highest). Which one of the following strategies would be the most beneficial for Vasil?
a. Build selectively on strengths
b. Upgrade product line.
c. Focus on attractive segments.
d. Avoid investments.
2.Topic A.1.e.
Which one of the following best describes tactical profit plans?
a. Detailed, short-term, broad responsibilities, qualitative.
b. Broad, short-term, responsibilities at all levels, quantitative.
c. Detailed, short-term, responsibilities at all levels, quantitative.
d. Broad, long-term, broad responsibilities, qualitativ
e.
3. Topic A.2.a.
Just-in-time production is also called
a. kaizen.
b. lean manufacturing.
c. activity-based management.
d. backflush costing.
4. Topic A.2.a.
Which of the following is not a correct comparison of a just-in-time system with a
traditional system?
Traditional Just-in-Time
a. Longer lead times Shorter lead times
b. Inventory is an asset Inventory is a liability
c. Some scrap tolerated Zero defects desired
d. Lot size based on immediate need Lot size based on formulas
5. Topic A.2.b.
Starr Company uses material requirements planning (MRP) and manufactures a product with the following product structure tree.
Starr has just received an order for 100 units of X, the finished product. The company has 20 units of X, 100 units of B, and 50 units of E in inventory. How many units of E must Starr purchase in order to fill the order?
a. 1,000.
b. 830.
c. 800.
d. 550.
Antlers, Inc. produces a single product that sells for $150 per unit. The product is processed through the Cutting and Finishing departments. Additional data for these departments are as follows. Cutting Finishing Annual capacity (36,000 direct labor hours
available in each department) 180,000 units 135,000 units Current production rate (annualized) 108,000 units 108,000 units Fixed manufacturing overhead $1,296,000 $1,944,000 Fixed selling and administrative expense $864,000 $1,296,000 Direct materials cost per unit $45 $15
The current production rate is the budgeted rate for the entire year. Direct labor employees earn $20 per hour and the company has a “no layoff” policy in effect. What is the amount of the throughput contribution per unit as computed using the theory of constraints? a. $90.00.
b. $76.67.
c. $46.67.
d. $26.67.
7. Topic A.2.c.
Three of the basic measurements used by the Theory of Constraints (TOC) are
a.
gross margin (or gross profit), return on assets, and total sales. b.
number of constraints (or subordinates), number of non-constraints, and operating leverage c.
throughput (or throughput contribution), inventory (or investments), and operational expense. d. fixed manufacturing overhead per unit, fixed general overhead per unit, and unit gross margin (or gross profit).
8. Topic B.1.c. Samsec Inc. has four major divisions, each doing business in a different market. Market growth rates and relative competitive positions for each of these four divisions are shown below. A
relative competitive position is defined as the division’s market share relative to the market share of the next largest competitor in the industry. Business growth rate Relative competitive position Division A 13% 4%
Division B 12% 110%
Division C 5% 12%
Division D 4% 170% Which division most likely does not have the strategic potential to generate a significant level of cash? a. Division A.
b. Division B.
c. Division C.
d. Division D.
Which one of the following is not an element of the strategic marketing process?
a. Customer segmentation.
b. Market selection.
c. Value positioning.
d. Product pricing.
10. Topic B.2.a.
In marketing research, basic decisions about whether to gather information from surveys or from observing customers' behaviors are part of which stage of the research process?
a. Defining the problem and research objectives.
b. Developing the research plan.
c. Collecting the information.
d. Analyzing the information.
11. Topic B.3.b.
Under which one of the following targeting strategies, does a firm attempt to serve all customer groups with all the products they might need?
a. Single-segment concentration.
b. Market specialization.
c. Full market coverage.
d. Product specialization.
12. Topic B.1.b.
SWOT analysis is concerned primarily with which of the following two factors?
a. The internal environment and the external environment.
b. Market share and market growth.
c. Product profitability and economic resources.
d. Customers’ and competitors’ profiles.
13. Topic B.3.a.
The process of dividing all potential consumers into smaller groups of buyers with distinct needs, characteristics, or behaviors, who might require a similar product or service mix, is called
a. strategic planning.
b. market segmentation.
c. product positioning.
d. objective setting.
In marketing toothbrushes as its single product, Hollow Company uses one marketing mix, believing all toothbrush users have similar needs. Such an approach to target marketing is called
a. mass marketing.
b. niche marketing.
c. micromarketing.
d. concentrated marketing.
15. Topic B.4.a.
Many retail stores have found that it is more profitable to sell a product carrying the store’s brand name as opposed to the more well-known brands. This is an example of a
a. manufacturer’s brand.
b. national brand.
c. private bran
d.
d. co-brand.
16. Topic C.1.c.
City Development, Inc. is considering a new investment project which will involve building a large office block in Frankfurt-am-Maine. The firm's financial analysis department has estimated that the proposed investment has the following estimated rate of return distribution.
Rate of Return Probability
(5%) 30%
10% 50%
20% 20%
Calculate the expected rate of return.
a. 5.5%.
b. 7.5%.
c. 10.5%.
d. 11.7%.
17. Topic C.1.d.
Within a financial risk management context, the term Value at Risk (VAR) is defined as the
a. maximum value a company can lose.
b. maximum loss within a certain time period at a given level of confidence.
c. worst possible outcome given the distribution of outcomes.
d. most likely negative outcom
e.
Russell Inc. is evaluating four independent investment proposals. The expected returns
and standard deviations for each of these proposals are presented below.
Investment Expected Standard
Proposal Returns Deviation
I 16% 10%
II 14% 10%
III 20% 11%
IV 22% 15%
Which one of the investment proposals has the least relative level of risk?
a. Investment I.
b. Investment II.
c. Investment III.
d. Investment IV.
19. Topic C.2.a.
If a bond sells at a premium, the
a. stated coupon rate must be less than the required market rate.
b. nominal rate must be less than the yield rate.
c. bond purchase price must be more than the fair market value of the bon
d.
d. stated coupon rate must be more than the required market rat
e.
20. Topic C.2.b.
During the most recent fiscal year, Dongata Industries earned net income after tax of $3,288,000. The company paid preferred share dividends of $488,000 and common share dividends of
$1,000,000. The current market price of Dongata's common shares is $56 per share and the shares are trading at a price/earnings ratio of 8. How many common shares does Dongata have outstanding?
a. 350,000.
b. 400,000.
c. 411,000.
d. 469,714.
21. Topic C.2.a.
From an investor's viewpoint, the least risky type of bond in which to invest is a (n)
a. debenture bond.
b. deep discount bond.
c. income bon
d.
d. secured bond.
Which one of the following events will most likely result in a higher price-earnings ratio for a company's common shares?
a. Investors' required rate of return on the common shares falls.
b. The rate of growth in dividends is expected to decline.
c. The economy is expected to enter a recession.
d. The dividend yield increases when the dividend per share remains unchanged.
23. Topic C.2.d.
Which one of the following is not a determinant in valuing a call option?
a. Exercise price.
b. Expiration date.
c. Forward contract price.
d. Interest rat
e.
24. Topic C.1.f.
A market analyst has estimated the equity beta of Modern Homes Inc. to be 1.4. This beta implies that the company’s
a. systematic risk is lower than that of the market portfolio.
b. systematic risk is higher than that of the market portfolio.
c. unsystematic risk is higher than that of the market portfolio.
d. total risk is higher than that of the market portfolio.
25. Topic C.1.f.
An analyst covering Guilderland Mining Co. common stock estimates the following information for next year.
Expected return on the market portfolio 12%
Expected return on Treasury securities 5%
Expected beta of Guilderland 2.2
Using the CAPM, the analyst’s estimate of next year’s risk premium for Guilderland’s stock is closest to
a. 7.0%.
b. 10.4%.
c. 15.4%.
d. 21.4%.
Hi-Tech Inc. has determined that it can minimize its weighted average cost of capital (WACC) by using a debt/equity ratio of 2/3. If the firm’s cost of debt is 9% before taxes, the cost of equity is estimated to be 12% before taxes, and the tax rate is 40%, what is the firm’s WACC?
a. 6.48%.
b. 7.92%.
c. 9.36%.
d. 10.80%.
27. Topic C.5.a.
The chief financial officer of Smith Glass Inc. follows the policy of matching the maturity of assets with the maturity of financing. The implications of this policy include all of the following, except that
a. the seasonal expansion of cash, receivables, and inventory should be financed by short-term
debt such as vendor payables and bank debt.
b. the minimum level of cash, receivables, and inventory required to stay in business can be
considered permanent, and financed with long-term debt or equity.
c. cash, receivables, and inventory should be financed with long-term debt or equity.
d. long-term assets, like plant and equipment, should be financed with long-term debt or equity.
28. Topic C.4.b.
A consultant recommends that a company hold funds for the following two reasons.
Reason #1: cash needs can fluctuate substantially throughout the year.
Reason #2: opportunities for buying at a discount may appear during the year.
The cash balances used to address the reasons given above are correctly classified as Reason #1 Reason #2
a. Speculative balances Speculative balances.
b. Speculative balances Precautionary balances.
c. Precautionary balances Speculative balances.
d. Precautionary balances Precautionary balances.
29. Topic C.4.d.
Yonder Motors sells 20,000 automobiles per year for $25,000 each. The firm’s average receivables are $30,000,000 and average inventory is $40,000,000. Yonder’s average collection period is closest to which one of the following? Assume a 365-day year.
a. 17 days.
b. 22 days.
c. 29 days.
d. 61 days.
An industry management technique designed to minimize inventory investment by having materials arrive at the time they are needed for use is known as
a. the economic order quantity model (EOQ).
b. material resource planning (MRP).
c. first-in first-out (FIFO).
d. just-in-time (JIT).
31. Topic C.5.a.
With respect to the use of commercial paper by an industrial firm, which one of the following statements is most likely to be true?
a. The commercial paper is issued through a bank.
b. The commercial paper has a maturity of 60-270 days.
c. The commercial paper is secured by the issuer’s assets.
d. The commercial paper issuer is a small company.
32. Topic C.5.b.
The Red Company has a revolving line of credit of $300,000 with a one-year maturity. The terms call for a 6% interest rate and a ?% commitment fee on the unused portion of the line of credit. The average loan balance during the year was $100,000. The annual cost of this financing arrangement is
a. $6,000.
b. $6,500.
c. $7,000.
d. $7,500.
33. Topic C.5.b.
Maple Motors buys axles in order to produce automobiles. Maple carries an average credit balance of $25,000,000 with its axle supplier. The axle supplier provides credit terms of 1/10 net 25. The nominal annual cost of Maple not taking the trade discount is closest to which one of the following? Assume a 360-day year.
a. 14.4%.
b. 14.5%.
c. 24.0%.
d. 24.2%.
Romashka, Inc. plans on introducing a new product. The marketing manager forecasts a unit selling price of $500. The variable cost per unit is estimated to be $100 per unit. In addition, there is a total of $110,000 fixed indirect manufacturing cost and $150,000 in fixed operating costs associated with these units.
What quantity will the company have to sell to break even?
a. 220 units.
b. 275 units.
c. 520 units.
d. 650 units.
35. Topic D.3.b.
Associated Supply, Inc. is considering introducing a new product that will require a $250,000 investment of capital. The necessary funds would be raised through a bank loan at an interest rate of 8%. The fixed operating costs associated with the product would be $122,500 while the contribution margin percentage would be 42%. Assuming a selling price of $15 per unit, determine the number of units (rounded to the nearest whole unit), Associated would have to sell to generate earnings before interest and taxes (EBIT) of 32% of the amount of capital invested in the new product.
a. 35,318 units.
b. 32,143 units.
c. 25,575 units.
d. 23,276 units.
Following are the operating results of the two segments of Parklin Corporation.
Segment A Segment B Total Sales $10,000 $15,000 $25,000
Variable costs of goods sold 4,000 8,500 12,500
Fixed costs of goods sold 1,500 2,500 4,000
Gross
margin 4,500 4,000 8,500
administrative
2,000 3,000 5,000
and
selling
Variable
Fixed selling and administrative 1,500 1,500 3,000
Operating income (loss) $ 1,000 $ (500) $ 500
Fixed costs of goods sold are allocated to each segment based on the number of employees. Fixed selling and administrative expenses are allocated equally. If Segment B is eliminated, $1,500 of fixed costs of goods sold would be eliminated. Assuming Segment B is closed, the effect on operating income would be
a. an increase of $500.
b. an increase of $2,000.
c. a decrease of $2,000.
d. a decrease of $2,500.
37. Topic D.5.c.
Pazer Inc. produces portable televisions. Pazer’s product manager proposes to increase the cost structure by adding voice-activated volume/channel controls to the television, and also adding three additional repair personnel to deal with products returned due to defects. Are these costs value-added or nonvalue-added?
Cost of Voice-activated Controls Cost of Additional Repair Personnel
a. Value-added Value-added
b. Value-added Nonvalue-added
c. Nonvalue-added Value-added
d. Nonvalue-added Nonvalue-added
38. Topic D.5.c.
Systematic evaluation of the trade-offs between product functionality and product cost while still satisfying customer needs is the definition of
a. activity-based management.
b. theory of constraints.
c. total quality management.
d. value engineering.
Stroylux Inc. is in the process of making an investment decision. It has identified the following alternative investment opportunities: real estate, stocks, bonds, or mutual funds. After determination of the investment alternatives, Stroylux will go thorough all of the following logical steps of the decision analysis model, except
a. obtaining information about the investment alternatives.
b. evaluating the investment alternatives.
c. selecting an alternative which best fits the investment criteria.
d. defining the problem of high unused cash balanc
e.
40. Topic D.2.a.
In the decision making process, differential cost is a(n)
a. sunk cost of alternative courses of action.
b. fixed cost of alternative courses of action.
c. opportunity cost of alternative courses of action.
d. cost that changes among alternative courses of action.
41. Topic D.1.a.
Okna Inc. has been experiencing sales difficulties over the last two years. Management has identified the problem: poor customer service and lack of internal coordination among the marketing department, the production department, and the finance department. In making a decision to solve the problem, management took the following logical steps (not necessarily in this order):
1.Concluded that extensive training would improve performance and efficiency of all the
involved departments; replacing the head of the marketing department will increase
profitability only; and investing in information technology will improve customer
satisfaction, but will not solve communication problems.
2.Identified three possible ways to solve the problem: extensive personnel training in the
involved departments; replacing the head of the marketing department; or investing in IT.
3.Agreed upon key performance indicators, such as customer satisfaction, customer retention,
and product and customer profitability to assess the effectiveness of the proposed solution.
4.Decided to conduct extensive training for marketing department personnel and limited
communication training for production and finance departments.
What is the correct sequence of these logical decision-making steps?
a. 1,2,3,4.
b. 2,1,3,4.
c. 3,1,2,4.
d. 2,3,1,4.
42. Topic D.3.a.
Specialty Cakes Inc. produces two types of cakes, a round cake and a heart-shaped cake. Total fixed costs for the firm are $92,000. Variable costs and sales data for these cakes are presented below.
Round Cake Heart-shape Cake Selling price per unit $12 $20
Variable cost per unit $8 $15
Budgeted sales (units) 10,000 15,000
How many cakes will be required to reach the breakeven point?
a. 8,000 round cakes and 12,000 heart-shaped cakes.
b. 9,000 round cakes and 11,000 heart-shaped cakes.
c. 10,000 round cakes and 10,000 heart-shaped cakes.
d. 23,000 round cakes and 18,400 heart-shaped cakes.
43. Topic D.4.a.
McCann Company can manufacture one of two special orders with their existing capacity. Special Order A is for 100,000 units and Special Order B is for 200,000 units. Cost and revenue data per unit are as follows.
Per Unit
Special order A B
price $.7000
$.4500
Sales
materials .4550 .2775
Direct
labor
Direct
Variable .1100 .0993
Fixed .0300 .0089
overhead
Manufacturing
Variable .0430 .0330
Fixed .0370 .0523 Variable marketing costs, already
incurred to obtain the order .0900 .0912 Fixed marketing and administrative costs .0950 .0878
Based on the above information, which one of the following statements correctly identifies the effect on pretax profit if the optimal decision is made?
a. $200 increase if Special Order A is taken.
b. $9,200 increase if Special Order A is taken.
c. $13,430 increase if Special Order A is taken.
d. $8,040 increase if Special Order B is taken.
Finn Products, a start-up company, wants to use cost-based pricing for its only product, a unique new video game. Finn expects to sell 10,000 units in the upcoming year. Variable costs will be $65 per unit and annual fixed operating costs (including depreciation) amount to $80,000. Finn’s balance sheet is as follows.
Assets Liabilities & Equity Current assets $100,000 Accounts payable $ 25,000
Plant & equipment 425,000 Debt 200,000
Equity 300,000
If Finn wants to earn a 20% return on equity, at what price should it sell the new product?
a. $75.00.
b. $78.60.
c. $79.00.
d. $81.00.
45. Topic E.1.d.
The Chief Financial Officer of Pauley Inc. has requested an evaluation of a proposed acquisition of a new machine at a purchase price of $60,000, and with installation costs of $10,000. A $3,000 increase in working capital will be required. The machine will have a useful life of four years after which it can be sold for $10,000. The estimated annual incremental operating revenues and cash operating expenses are $150,000 and $100,000, respectively, for each of the four years. Pauley's effective income tax rate is 40% and the cost of capital is 12%. Pauley uses straight-line depreciation for both financial reporting and income tax purposes.
Pauley's estimated after-tax cash flow in the fourth year, at which time the equipment will be sold, will be
a. $34,000.
b. $45,000.
c. $46,000.
d. $49,000.
The management of Pelican Inc. is evaluating a proposed acquisition of a new machine at a purchase price of $180,000, and with installation costs of $10,000. A $9,000 increase in working capital will be required. The machine will have a useful life of four years after which it can be sold for $30,000. The estimated annual incremental operating revenues and cash operating expenses are $450,000 and $300,000, respectively, for each of the four years. Pelican’s effective income tax rate is 40% and the cost of capital is 12%. Pelican uses straight-line depreciation for both financial reporting and income tax purposes.
If the project is accepted, the estimated incremental after-tax operating cash flows at the end of the first year will be
a. $ 99,000.
b. $105,000.
c. $108,000.
d. $150,000.
47. Topic E.1.c.
Mobile Home Manufacturing Inc. is evaluating a proposed acquisition of a new machine at a purchase price of $380,000 and installation charges that will amount to $20,000. A $15,000 increase in working capital will be required. The machine will have a useful life of four years after which it can be sold for $50,000. The estimated annual incremental operating revenues and cash operating expenses are $750,000 and $500,000, respectively, for each of the four years. Mobile Home’s tax rate is 40% and the cost of capital is 12%. Mobile Home uses straight-line depreciation for both financial reporting and income tax purposes.
If Mobile Home accepts the project the initial investment will be
a. $350,000.
b. $365,000.
c. $385,000.
d. $415,000.
A company is considering the purchase of a new machine to replace a five-year old machine and has gathered the following information.
new
$50,000
the
machine
Purchase
price
of
Installation cost of the new machine 4,000
Market value (selling price) of the old machine 5,000
Book value of the old machine 2,000
Increase in net working capital if new machine is installed 1,000
Effective income tax rate 40%
If the company replaces the old machine with the new machine, what is the cash flow in period 0?
a. ($49,000).
b. ($51,200).
c. ($51,800).
d. ($53,000).
49. Topic E.2.a.
Wilkinson Inc., which has a cost of capital of 12%, invested in a project with an internal rate of return (IRR) of 14%. The project is expected to have a useful life of four years and it will produce net cash inflows as follows.
Year Net Cash Inflows
1 $1,000
2 2,000
3 4,000
4 4,000
The initial cost of this project amounted to
a. $7,483.
b. $9,647.
c. $11,000.
d. $12,540.
Gibber Corporation has an opportunity to sell a newly developed product in the United States for a period of five years. The product license would be purchased from NewGroup Company. Gibber would be responsible for all distribution and product promotion costs. NewGroup has the option to renew the agreement, with modifications, at the end of the initial five-year term. Gibber has developed the following estimated revenues and costs that would be associated with the new product.
required $
equipment
120,000
new
Cost
of
capital
required 200,000
Additional
working
Salvage value of equipment in year 5 20,000
Annual revenues and costs
revenues 400,000
Sales
Costs of goods sold 250,000
Out-of-pocket operating cost 70,000
The working capital required to support the new product would be released for investment elsewhere if the product licensing agreement is not renewed.
Using the net present value method of analysis, and ignoring income tax, the net present value of this product agreement assuming Gibber has a 20% cost of capital would be
a. $7,720.
b. ($64,064).
c. ($72,680).
d. ($127,320).
51. Topic E.2.a.
Bennet Inc. uses the net present value method to evaluate capital projects. Bennet's required rate of return is 10%. Bennet is considering two mutually exclusive projects for its manufacturing business. Both projects require an initial outlay of $120,000 and are expected to have a useful life of four years. The projected after-tax cash flows associated with these projects are as follows.
Year Project X Project Y
1 $40,000 $10,000
2 40,000 20,000
3 40,000 60,000
4 40,000 80,000
Total $160,000 $170,000
Assuming adequate funds are available, which of the following project options would you recommend that Bennet's management undertake?
a. Project X only.
b. Project Y only.
c. Projects X and Y.
d. Neither project.
Brown and Company uses the internal rate of return (IRR) method to evaluate capital projects. Brown in considering four independent projects with the following IRRs.
Projects IRR
10%
I
II
12%
14%
III
15%
IV
Brown's cost of capital is 13%. Which one of the following project options should Brown accept based on IRR?
a. Projects I and II only.
b. Projects III and IV only.
c. Project IV only.
d. Projects I, II, III, and IV.
53. Topic E.4.c.
Woods Inc. is considering four independent investment proposals. Woods has $3 million available for investment during the present period. The investment outlay for each project and its projected net present value (NPV) is presented below.
Project Investment Cost NPV
I $ 500,000 $ 40,000
II 900,000 120,000
III 1,200,000 180,000
IV 1,600,000 150,000
Which of the following project options should be recommended to Woods' management?
a. Projects I, II, and III only.
b. Projects I, II, and IV only.
c. Projects II, III, and IV only.
d. Projects III and IV only.
一 单项选择题(每小题3分,共15分) 1.设lim ()x a f x k →=,那么点x =a 是f (x )的( ). ①连续点 ②可去间断点 ③跳跃间断点 ④以上结论都不对 2.设f (x )在点x =a 处可导,那么0()(2)lim h f a h f a h h →+--=( ). ①3()f a ' ②2()f a ' ③()f a ' ④ 1()3f a ' 3.设函数f (x )的定义域为[-1,1],则复合函数f (sinx )的定义域为( ). ①(-1,1) ②,22ππ??-??? ? ③(0,+∞) ④(-∞,+∞) 4.设2()()lim 1() x a f x f a x a →-=-,那么f (x )在a 处( ). ①导数存在,但()0f a '≠ ②取得极大值 ③取得极小值 ④导数不存在 5.已知0lim ()0x x f x →=及( ),则0 lim ()()0x x f x g x →=. ①g (x )为任意函数时 ②当g (x )为有界函数时 ③仅当0lim ()0x x g x →=时 ④仅当0 lim ()x x g x →存在时 二 填空题(每小题5分,共15分) 1.sin lim sin x x x x x →∞-=+____________. 2.31lim(1)x x x +→∞+=____________. 3.()f x =那么左导数(0)f -'=____________,右导数(0)f +'=____________. 三 计算题(1-4题各5分,5-6题各10分,共40分) 1.111lim()ln 1 x x x →-- 2.t t x e y te ?=?=?,求22d y dx 3.ln(y x =,求dy 和22d y dx . 4.由方程0x y e xy +-=确定隐函数y =f (x ) ,求dy dx . 5.设111 1,11n n n x x x x --==++,求lim n x x →∞.
浙江工商大学2014/2015学年第一学期期末考试卷A 课程名称:微积分(上)A 层 考试方式: 闭 卷 完成时限: 120分钟 班级名称: 学 号: 姓 名:___________ 一、填空题（每小题3分,共15分） 1.函数()1,1f x x =+则()()f f x 的定义域是 . 2.点0=x 为函数e ,0,()ln(1),10 x x f x x x -?>=?+-<≤?的第 类间断点. 3.若函数x y sin =,则=)2015(y . 4.()sin 2d 3x = . 二、选择题（每小题3分,共15分） 1.当0→x 时,与x 等价的无穷小是( ). A.x x +3 1-
C.)1e sin(-x D.x cos 1- 2.下列函数中,在点0=x 处可导的是( ). A.||)(x x x f = B.|sin |)(x x f = C.?????=≠=0,0,0,1sin )(x x x x x f D.???>≤+=0, ,0,1)(2x x x x x f 3. 设()x f x x =，则其导数为( ). A. x x x f =')( B. x x x f x ln )(=' C. )1(ln )(+='x x x f x D. 1)(-='x x x f 4.设)(x f 的导数在a x =处连续,又1)(lim -=-'→a x x f a x ,则( ). A.a x =是)(x f 的极小值点 B.a x =是)(x f 的极大值点 C.))(,(a f a 是曲线)(x f 的拐点 D.a x =不是)(x f 的极值点,))(,(a f a 也不是曲线)(x f 的拐点 5.下列等式中,正确的是( ). A.)(d )(x f x x f ?=' B.)()(d x f x f ?= C.)(d )(d d x f x x f x ?= D.)(d )(d x f x x f ?= 三、计算题（写出必要的解题步骤,每小题6分,共48分） 1.求极限()()1sin 0lim 12x x f x →-，其中()()00,02f f '==，当0x ≠时()0f x ≠.
(3343)．微分方程0cos tan =-+'x x y y 的通解为 x C x y cos )(+=。 (4455)．过点)0,2 1(且满足关系式11arcsin 2 =-+ 'x y x y 的曲线方程为 21arcsin - =x x y 。 (4507)．微分方程03='+''y y x 的通解为 2 2 1x C C y + =。 (4508)．设)(),(),(321x y x y x y 是线性微分方程)()()(x f y x b y x a y =+'+''的三个特解，且 C x y x y x y x y ≠--) ()() ()(1312，则该微分方程的通解为 )())()((())()((1132121x y x y x y C x y x y C y +-+-=。 (3081)．设x e x y x y -++=+=22213,3是某二阶线性非齐次微分方程的两个特解，且相 应齐次方程的一个解为x y =3，则该微分方程的通解为x e C x C x y -+++=212 3。 (4725)．设出微分方程x e xe x y y y x x 2cos 32++=-'-''-的一个特解形式 )2sin 2cos ()(*x F x E e e D Cx x B Ax y x x +++++=-。 (4476)．微分方程x e y y y =+'-''22的通解为 )sin cos 1(21x C x C e y x ++=。 (4474)．微分方程x e y y 24=-''的通解为 x x e x C e C y 222141??? ? ? ++=-。 (4477)．函数x C x C y 2s i n 2c o s 21+=满足的二阶线性常系数齐次微分方程为04=+''y y 。 (4532)．若连续函数)(x f 满足关系式 2ln )2 ()(20 +=? x dt t f x f ，则=)(x f 2ln 2x e 。 (6808)．设曲线积分 ?--L x ydy x f ydx e x f cos )(sin ])([与路径无关，其中)(x f 具有一阶 连续导数，且0)0(=f ，则)(x f 等于[ ] (A) )(2 1x x e e --。 (B) )(21 x x e e --。
微积分(上)期末考试试题（B）
对外经济贸易大学 2003-2004学年第一学期 《微积分》（上）期末考试试卷(B) 课程课序号CMP101??（1~14） 学号：___________ 姓名：___________ 班级：___________ 成绩：___________ 题号 一 二 三 四 五 六 总分 成 绩 一、 选择题 (选出每小题的正确答案，每小题２分，共计８分) 1． 下列极限正确的是 _________。 （A ）1 0lim 20x x + →= （B ） 10lim 20 x x - →= （C ）1lim(1) x x e x →∞ -=- （D ） 01lim (1)1x x x +→+= 2．若()(),f x x a x x φφφ=-≠其中（）为连续函数，且（a ）0，() f x 在 x a =点_________。 （A ） 不连续 （B ） 连续 （C ）可导 （D ） 不可导
３． 设f （x ）有二阶连续导数，且 2 () (0)0,lim 1,_______x f x f x →'''==则。 () 0()A x f x =是的极大值点 ()0(0)B f （，）是f(x)的拐点 ()0()C x f x =是的极小值点 ())0D f x x =（在处是否取极值不确定 ４．下列函数中满足罗尔定理条件的是 。 ()ln(2) [0,1] A f x x x =-（） 2 01()0 1 x x B f x x ?≤<=? =?（） ()sin sin [0,] C f x x x x π=+（） 2 1 ()1[1,1] D f x x =- -（） 5．若()(),f x x φ''=则下列各式 成立。 () ()()0A f x x φ-= () ()()B f x x C φ-= () ()()C d f x d x φ=?? () ()()d d D f x dx x dx dx dx φ=?? 二、 填空题（每小题3分，共18分） 1. 设0 (2) ()0(0)0,lim 1sin x f x f x x f x →-===-在处可导，且，那么曲线() y f x =在原点处的切线方程是__________。 ２．设函数f （x ）可导，则2 (4)(2)lim 2 x f x f x →--=-_________。 3．设ln ,()x xf x dx x '=?为f(x)的一个原函数那么 。 4 ． 设 2121,2ln 3x x y a x bx x a b ===++均是的极值点，则、的值为 。 5． 设某商品的需求量Ｑ是价格Ｐ的函数
西南大学课程考核
《高等数学IA 》课程试题 【A 】卷 (1) The function 4 14 )(-= x x f at x = 4 is ( ). A. not continuous, f (4) does not exist and )(lim 4 x f x → does not exist. B. continuous. C. not continuous, )(lim 4 x f x → exists but f (4) does not exist D. not continuous, )(lim 4 x f x → and f (4) exist but )4()(lim 4 f x f x ≠→. (2) For the function y = arcsin x , we have the assert ( ). A ．'y is undefined at x = -1 and x = 1, so its graph has not tangent lines at ??? ??2π, 1 and ??? ? ? --2π,1. B ．since its graph has not tangent lines at ??? ??2π, 1 and ??? ? ? --2π,1,'y is undefined at x = -1 and x = 1. C ．'y is defined at x = -1 and x = 1, and its graph has tangent lines at ??? ??2π, 1 and ??? ?? --2π,1. D ．'y is undefined at x = -1 and x = 1, and its graph has tangent lines at ?? ? ??2π, 1 and ??? ? ?--2π,1. (3) =?x x x d )(ln 1 5( ) . A. C x x +- 4 )(ln 41 B. C x +-6)(ln 61. C. C x +- 4)(ln 41 D. C x x +-6 ) (ln 61 . (4) The definite integral =+?-x x x d 131 1 32 ( ). A. 334 B. 324. C. 423 D. 4 33 (5) Area of shaded region in the following figure is ( ).
微积分期末测试题及答 案 Document number：NOCG-YUNOO-BUYTT-UU986-1986UT
一 单项选择题(每小题3分,共15分) 1.设lim ()x a f x k →=,那么点x =a 是f (x )的( ). ①连续点 ②可去间断点 ③跳跃间断点 ④以上结论都不对 2.设f (x )在点x =a 处可导,那么0()(2)lim h f a h f a h h →+--=( ). ①3()f a ' ②2()f a ' ③()f a ' ④1()3f a ' 3.设函数f (x )的定义域为[-1,1],则复合函数f (sinx )的定义域为( ). ①(-1,1) ②,22ππ??-???? ③(0,+∞) ④(-∞,+∞) 4.设2 ()()lim 1()x a f x f a x a →-=-,那么f (x )在a 处( ). ①导数存在,但()0f a '≠ ②取得极大值 ③取得极小值 ④导数不存在 5.已知0lim ()0x x f x →=及( ),则0 lim ()()0x x f x g x →=. ①g (x )为任意函数时 ②当g (x )为有界函数时 ③仅当0lim ()0x x g x →=时 ④仅当0 lim ()x x g x →存在时 二 填空题(每小题5分,共15分) sin lim sin x x x x x →∞-=+. 31lim(1)x x x +→∞+=. 3.()f x =那么左导数(0)f -'=____________,右导数(0)f +'=____________. 三 计算题(1-4题各5分,5-6题各10分,共40分) 1.111lim()ln 1 x x x →-- 2.t t x e y te ?=?=? ,求22d y dx 3.ln(y x =,求dy 和22d y dx . 4.由方程0x y e xy +-=确定隐函数y =f (x ) ,求 dy dx . 5.设111 1,11n n n x x x x --==+ +,求lim n x x →∞.
高学试题及答案 选择题（本大题共 40 小题，每小题 2.5 分，共 100 分） 1．设 f(x)=lnx ，且函数 (x) 的反函数 1 (x)= 2(x+1) ，则 f (x) （ B ） x-2 x+2 2-x x-1 x+2 ln ln ln ln A. x+2 B. x-2 C. x+2 D. 2-x e t 2 dt 2． lim e t x 1 cosx （ A ） x 0 A ． 0 B ． 1 C ．-1 D ． 3．设 y f ( x 0 x) f ( x 0 ) 且函数 f (x) 在 x x 0 处可导，则必有（ A ） A. lim y 0 B. y 0 C.dy 0 D. y dy x 0 4．设函数 f(x)= 2x 2 , x 1 ，则 f(x) 在点 x=1处（ C ） 3x 1,x 1 A. 不连续 B. 连续但左、右导数不存在 C. 连续但不可导D. 可导 5．设 xf(x)dx=e -x 2 C ，则 f(x)= （ D ） A.xe 6. 设 I -x 2 B.-xe -x 2 C.2e -x 2 D.-2e -x 2 ( x 2 y 2 ) dxdy ，其中 D 由 x 2 y 2 a 2 所围成，则 I =( B ). D (A) 2 a 2 rdr a 4 (B) 2 a 2 rdr 1 a 4 d a d r 2 2 a 2 dr 2 a 3 2 a 2 adr 2 a 4 (C) d r (D) d a 3 7. 若 L 是上半椭圆 x a cost , ydx xdy 的值为 ( C ). y 取顺时针方向 , 则 b sin t , L (A)0(B) ab (C) ab (D) 2 8. 设 a 为非零常数 , 则当 ( B ) 时 , 级数 a 收敛 . n 1 r n ab (A) | r | | a | (B) | r | | a | (C) | r | 1 (D) | r | 1 9. lim u n 0 是级数 u n 收敛的 ( D ) 条件 . n n 1 (A) 充分 (B) 必要 (C) 充分且必要 (D) 既非充分又非必要
一、 选择题 (选出每小题的正确选项，每小题２分，共计10分) 1．1 lim 2x x - →=_________。 （A ） - （B ） + （C ） 0 （D ） 不存在 2．当0x →时，()x x f x x += 的极限为 _________。 （A ） 0 （B ） 1 （C ）2 （D ） 不存在 ３． 下列极限存在，则成立的是_________。 0()()() lim ()x f a x f a A f a x - ?→+?-'=?0()(0) ()lim (0) x f tx f B tf x →-'= 0000()()()lim 2()t f x t f x t C f x t →+--'= 0()() ()lim ()x f x f a D f a a x →-'=- ４． 设f （x ）有二阶连续导数，且()0 () (0)0,lim 1,0()_______x f x f f f x x →'''==则是的。 （A ） 极小值 （B ）极大值（ C ）拐点 （D ） 不是极值点也不是拐点 5．若()(),f x g x ''=则下列各式 成立。 ()()()0A f x x φ-=()()()B f x x C φ-= () ()() C d f x d x φ= ?? () ()()d d D f x dx x dx dx dx φ=?? 二、 填空题（每小题3分，共18分） 1. 设0 (2) ()0(0)0,lim 1sin x f x f x x f x →===-在处可导，且，那么曲线()y f x =在原点处的切线方程是__________。 ２．函数()f x =[0，3]上满足罗尔定理，则定理中的= 。 3．设1 (),()ln f x f x dx x '=?的一个原函数是 那么 。 4．设(),x f x xe -=那么2阶导函数 ()___f x x ''=在点取得极_____值。 5．设某商品的需求量Ｑ是价格Ｐ的函数5Q =-，那么在Ｐ＝４的水平上，若价格 下降1％，需求量将 。 6．若,1 1),(+-= =x x u u f y 且,1)('u u f =dy dx = 。 三、计算题（每小题6分，共42分）： 1、 求 11ln (ln ) lim x x e x -→
中南民族大学06、07微积分（下）试卷 及参考答案 06年A 卷 评分 阅卷人 1、已知22 (,)y f x y x y x +=-,则=),(y x f _____________. 2、已知,则= ?∞ +--dx e x x 0 21 ___________. π =? ∞ +∞ --dx e x 2 3、函数 22 (,)1f x y x xy y y =++-+在__________点取得极值. 4、已知y y x x y x f arctan )arctan (),(++=,则= ')0,1(x f ________. 5、以x e x C C y 321)(+=(21,C C 为任意常数)为通解的微分方程是 ____________________. 二、选择题(每小题3分,共15分) 评分 阅卷人 6 知dx e x p ?∞ +- 0 )1(与?-e p x x dx 1 1 ln 均收敛, 则常数p 的取值范围是( ). (A) 1p > (B) 1p < (C) 12p << (D) 2p >
7 数???? ?=+≠++=0 ,0 0 ,4),(222 22 2y x y x y x x y x f 在原点间断, 是因为该函数( ). (A) 在原点无定义 (B) 在原点二重极限不存在 (C) 在原点有二重极限,但无定义 (D) 在原点二重极限存在,但不等于函数值 8、若 2 2223 11 1x y I x y dxdy +≤= --?? ,22223212 1x y I x y dxdy ≤+≤=--??, 2 2223 324 1x y I x y dxdy ≤+≤=--?? ,则下列关系式成立的是( ). (A) 123I I I >> (B) 213I I I >> (C) 123I I I << (D) 213I I I << 9、方程x e x y y y 3)1(596+=+'-''具有特解( ). (A) b ax y += (B) x e b ax y 3)(+= (C) x e bx ax y 32)(+= (D) x e bx ax y 323)(+= 10、设∑∞ =12n n a 收敛，则∑∞ =-1) 1(n n n a ( ). (A) 绝对收敛 (B) 条件收敛 (C) 发散 (D) 不定 三、计算题(每小题6分,共60分) 评分 评分 评阅人 11、求由2 3x y =,4=x ,0=y 所围图形绕y 轴旋转的旋转体的体积.
一元微积分期中考试答案 一． 填空题（每空3分，共15题） 1. e 1 2。21 3. 31 4。3 4 5. 1 6．第一类间断点 7。()dx x x x ln 1+ 8。 22sin(1)2cos(1)x x x e ++ 9。 0 10。11?????? ?+x e x 11．x x ne xe + 12。13 13。0 14。)1(223 +? =x y 15． 13y x =+ 二． 计算题 1. 解：,)(lim ,0)(lim 00b x f x f x x ==+?→→故0=b 。 …………………3分 a x f x f f x =?=′? →?)0()(lim )0(0 …………………3分 1)0()(lim )0(0=?=′+→+x f x f f x …………………3分 1=a 故当1=a ，0=b 时，)(x f 在),(+∞?∞内可导。 …………………1分 2. 解：=?+∞→])arctan ln[(lim ln /12x x x πx x x ln )arctan ln(lim 2?+∞→π = x x x x /1arctan ) 1/(1lim 22?+?+∞→π …………罗比达法则…………4分 =x x x x arctan )1/(lim 2+?++∞→π = )1/(1)1/()1(lim 2222x x x x ++?+∞→ = 2211lim x x x +?+∞→ = 1? ………………………4分 所以，原极限=1?e ………………………………………………………………………2分 3． 解：)'1)((''y y x f y ++= ，故 1) ('11)('1)(''?+?=+?+=y x f y x f y x f y ；……4分 3 2)]('1[)('')]('1[)'1)((''''y x f y x f y x f y y x f y +?+=+?++= …………………………………………6分 4.解：
精品文档 精品文档 ---○---○--- ---○---○--- ………… 评卷密封线………… 密封线内不要答题，密封线外不准填写考生信息，违者考试成绩按0分处理…………评卷密封 线………… 中南大学考试试卷 2009 ~2010学年 一 学期 微积分A 课程 （时间：10年1月21日，星期四，15：20—15：00，共计：100分钟） 一、填空题（本大题共5小题，每小题3分，总计15分） 1.])2(sin 11sin [lim x x x x x x x x +++∞ →= . 2. 函数32 y ax bx cx d =+++满足条件 时, 这函数没有极值. 3. 广义积分 =-+∞? dx e x 20 . 4.幂级数 n n n x n 30 212∑∞ =-的收敛半径=R ，收敛区间为 ． 5．曲线?? ???==++11 222z z y x 的参数方程为 .
精品文档 二、选择题（在每个小题四个备选答案中选出一个正确答案，填在括号中，本大题共5小题，每小题3分，总计15分） 1．当0→x 时，下列变量是无穷小量的是（ ）． （A ）x 1sin ； （B ）x e 1 ； （C ）)1ln(2x +； （D ）x e ． 2．设x e x f -=)(，则 ='? dx x x f ) (ln （ ） ． （A ）C x +- 1； （B ）C x x +ln 1； （C ） C x +1； （ D ）C e x x +1． 3. 若)(x f 是奇函数且)0(f '存在，则0=x 点是函数x x f x F ) ()(= 的（ ）． （A ）无穷间断点； （B ）可去间断点； （C ）连续点； （D ）振荡间断点． 4.如果b a ,是方程0)(=x f 的两个根，)(x f 在],[b a 上连续，在),(b a 内可导，那么方程 0)(='x f 在),(b a 内（ ） ． (A)只有一个根； (B)至少有一个根； (C)没有根； (D)以上结论都不对． 5．无穷级数 ∑ ∞ =--1 1)1(n p n n ，(0>p )敛散性是（ ）． (A)一定绝对收敛； (B)一定条件收敛； (C)一定收敛； (D)以上结论都不对．
高等数学(下册)期末考试试题 考试日期：2012年 院（系）别 班级 学号姓名 成绩 一、填空题：（本题共5小题，每小题4分，满分20分，把答案直接填在题中横线上） 1、已知向量a 、b 满足0a b +=，2a =，2b =，则a b ?=-4． 2、设ln()z x xy =，则32 z x y ?=??-(1/y2)． 3、曲面2 2 9x y z ++=在点(1,2,4)处的切平面方程为 2 (x-1)+4(y-2)+z-4=0． 4、设()f x 是周期为2π的周期函数，它在[,)ππ-上的表达式为()f x x =，则()f x 的傅里叶级数 在3x =处收敛于，在x π=处收敛于． 5、设L 为连接(1,0) 与(0,1)两点的直线段，则 ()L x y ds +=?√2． ※以下各题在答题纸上作答，答题时必须写出详细的解答过程，并在每张答题纸写上：姓名、学号、班级． 二、解下列各题：（本题共5小题，每小题7分，满分35分） 1、求曲线222 222 239 3x y z z x y ?++=??=+??在点0M (1,1,2)-处的切线及法平面方程． 2、求由曲面2222z x y =+及22 6z x y =--所围成的立体体积． 故所求的体积为V dv Ω =???22 2620 20 2(63)6d d dz d πρρθρπρρπ-==-=?? (7) 3、判定级数 1 1 (1) ln n n n n ∞ =+-∑是否收敛？如果是收敛的，是绝对收敛还是条件收敛？ 4、设(,)sin x z f xy y y =+，其中f 具有二阶连续偏导数，求2,z z x x y ?????．
第一学期期末高等数学试卷 一、解答下列各题 (本大题共16小题，总计80分) 1、(本小题5分) 求极限 lim x x x x x x →-+-+-2332121629124 2、(本小题5分) .d )1(22x x x ?+求 3、(本小题5分) 求极限lim arctan arcsin x x x →∞?1 4、(本小题5分) ? -.d 1x x x 求 5、(本小题5分) ．求dt t dx d x ?+2 021 6、(本小题5分) ??.d csc cot 46x x x 求 7、(本小题5分) ．求?ππ 2 1 21cos 1dx x x 8、(本小题5分) 设确定了函数求．x e t y e t y y x dy dx t t ==?????=cos sin (),22 9、(本小题5分) ． 求dx x x ?+3 01 10、(本小题5分) 求函数 的单调区间y x x =+-422 11、(本小题5分) ．求? π +2 02sin 8sin dx x x 12、(本小题5分) ．，求设 dx t t e t x kt )sin 4cos 3()(ωω+=- 13、(本小题5分) 设函数由方程所确定求．y y x y y x dy dx =+=()ln ,226 14、(本小题5分) 求函数的极值y e e x x =+-2 15、(本小题5分) 求极限lim ()()()()()()x x x x x x x →∞++++++++--121311011011112222 16、(本小题5分) .d cos sin 12cos x x x x ? +求 二、解答下列各题 (本大题共2小题，总计14分)
第一学期期末考试试卷 一、填空题(将正确答案写在答题纸的相应位置. 答错或未答，该题不得分.每小题3分，共15分.） 1. =→x x x 1 sin lim 0___0_____. 2. 设1 )1(lim )(2+-=∞→nx x n x f n ，则)(x f 的间断点是___x=0_____. 3. 已知(1)2f =，4 1 )1('-=f ,则 12 ()x df x dx -== _______. 4. ()a x x '=_______. 5. 函数434)(x x x f -=的极大值点为________. 二、单项选择题（从下列各题四个备选答案中选出一个正确答案，并将其代码写 在答题纸的相应位置.答案选错或未选者，该题不得分.每小题3分，共15分.） 1. 设)(x f 的定义域为)2,1(, 则)(lg x f 的定义域为________. A.)2lg ,0( B. ]2lg ,0[ C. )100,10( D.)2,1(. 2. 设对任意的x ，总有)()()(x g x f x ≤≤?，使lim[()()]0x g x x ?→∞ -=，则 lim ()x f x →∞ ______. A.存在且一定等于零 B. 存在但不一定等于零 C.不一定存在 D. 一定存在. 3. 极限=-→x x x x e 21lim 0________. A. 2e B. 2-e C. e D.不存在. 4. 设0)0(=f ，1)0(='f ，则=-+→x x f x f x tan ) 2()3(lim 0________. A.0 B. 1 C. 2 D. 5. 5. 曲线2 21x y x =-渐近线的条数为________. A ．0 B ．1 C ．2 D ．3. 三、（请写出主要计算步骤及结果，8分.） 求2 0sin 1lim sin x x e x x →--. 四、（请写出主要计算步骤及结果，8分.）
《高等数学》试题30 考试日期：2004年7月14日 星期三 考试时间：120 分钟 一.选择题 1. 当0→x 时，)1ln(x y +=与下列那个函数不是等价的 （ ） A)、x y = B)、x y sin = C)、x y cos 1-= D)、1-=x e y 2. 函数f(x)在点x 0极限存在是函数在该点连续的（ ） A )、必要条件 B )、充分条件 C )、充要条件 D )、无关条件 3. 下列各组函数中，)(x f 和)(x g 不是同一函数的原函数的有（ ）. A)、()()() 222 1 ,21)(x x x x e e x g e e x f ---=-= B) 、(( )) ()ln ,ln f x x g x x ==- C)、()()x x g x x f --=-=1arcsin 23,12arcsin )( D)、()2 tan ,sec csc )(x x g x x x f =+= 4. 下列各式正确的是（ ） A ）、2ln 2x x x dx C =+? B ）、sin cos tdt t C =-+? C ）、 2arctan 1dx dx x x =+? D ）、2 11 ()dx C x x -=-+? 5. 下列等式不正确的是（ ）. A ）、 ()()x f dx x f dx d b a =??????? B ）、()()()[]()x b x b f dt x f dx d x b a '=??????? C ）、()()x f dx x f dx d x a =??????? D ）、()()x F dt t F dx d x a '=???? ??'? 6. 0 ln(1)lim x x t dt x →+=?（ ） A ）、0 B ）、1 C ）、2 D ）、4 7. 设bx x f sin )(=，则=''?dx x f x )(（ ） A ）、 C bx bx b x +-sin cos B ） 、C bx bx b x +-cos cos C ）、C bx bx bx +-sin cos D ）、C bx b bx bx +-cos sin
微积分期末试卷 一、选择题（6×２） cos sin 1.()2,()()22 ()()B ()()D x x f x g x f x g x f x g x C π ==1设在区间（0，）内（ ）。 Ａ是增函数，是减函数是减函数，是增函数二者都是增函数二者都是减函数 2x 1 n n n n 20cos sin 1n A X (1) B X sin 21C X (1) x n e x x n a D a π →-=--==>、x 时，与相比是（ ） Ａ高阶无穷小 Ｂ低阶无穷小 Ｃ等价无穷小 Ｄ同阶但不等价无价小３、ｘ=０是函数ｙ=（１-sinx)的（ ） Ａ连续点 Ｂ可去间断点 Ｃ跳跃间断点 Ｄ无穷型间断点４、下列数列有极限并且极限为１的选项为（ ）n 1 X cos n = 2 00000001 () 5"()() ()()0''( )<0 D ''()'()06x f x X X o B X o C X X X X y xe =<===、若在处取得最大值，则必有（ ）Ａf ＇f ＇f ＇且f f 不存在或f 、曲线（ ） Ａ仅有水平渐近线 Ｂ仅有铅直渐近线Ｃ既有铅直又有水平渐近线 Ｄ既有铅直渐近线 1~6 DDBDBD 二、填空题
1 d 1 2 lim2,, x d x ax b a b → ++ = ｘ ｘ ２ ２ １ １、（ ）＝ ｘ+1 、求过点（２，０）的一条直线，使它与曲线ｙ＝相切。这条直线方程为： ｘ ２ ３、函数ｙ＝的反函数及其定义域与值域分别是： ２＋１ ｘ ５、若则的值分别为： ｘ＋2x-3 1In1 x+ ; 2 32 2 y x x =-; 3 2 log,(0,1), 1 x y R x = - ; 4(0,0) 5解：原式=11 (1)()1m lim lim2 (1)(3)34 77,6 x x x x m x m x x x m b a →→ -+++ === -++ ∴=∴=-= 三、判断题 1、无穷多个无穷小的和是无穷小（） 2、 sin lim x x x → -∞+∞ 在区间（，）是连续函数（） 3、 f"(x）=0一定为f(x)的拐点（） 4、若f(X)在 x处取得极值，则必有f(x)在 x处连续不可导（） 5、设函数ｆ(x)在[] 0,1上二阶可导且'()0A'0B'(1),(1)(0),A>B>C( ) f x f f C f f <===- 令（），则必有 1~5 FFFFT 四、计算题 1用洛必达法则求极限2 1 2 lim x x x e →
1、 (1) sin 2lim x x x →∞= 0 . (2) d(arctan )x = 2 1 d 1+x x (3) 21 d sin x x =? -cot +C x x (4).2() ()x n e = 22n x e . (5) x =? 26/3 2、 (6) The right proposition in the following propositions is ___A_____. A. If lim ()x a f x →exists and lim ()x a g x →does not exist then lim(()())x a f x g x →+does not exist. B. If lim ()x a f x →,lim ()x a g x →do bot h not exist then lim(()())x a f x g x →+does not exist. C. If lim ()x a f x →exists and lim ()x a g x →does not exist then lim ()()x a f x g x →does not exist. D. If lim ()x a f x →exists and lim ()x a g x →does not exist then () lim () x a f x g x →does not exist. (7) The right proposition in the following propositions is __B______. A. If lim ()()x a f x f a →=then ()f a 'exists. B. If lim ()()x a f x f a →≠ then ()f a 'does not exist. C. If ()f a 'does not exist then lim ()()x a f x f a →≠. D. If ()f a 'does not exist then the cure ()y f x =does not have tangent at (,())a f a . (8) The right statement in the following statements is ___D_____. A. sin lim 1x x x →∞= B. 1 lim(1)x x x e →∞+= C. 11d 1x x x C ααα += ++? D. 5511 d d 11b b a a x y x y =++?? (9) For continuous function ()f x , th e erroneous expression in the following expressions is ____D__. A. d (()d )()d b a f x x f b b =? B. d (()d )()d b a f x x f a a =-?
微积分期末试卷 选择题(6X2) 1?设f(x) 2cosx,g(x) (1严在区间(0,—)内()。 2 2 A f (x)是增函数，g (x)是减函数 Bf (x)是减函数，g(x)是增函数 C二者都是增函数 D二者都是减函数 2、x 0时，e2x cosx与sinx相比是() A高阶无穷小E低阶无穷小C等价无穷小D同阶但不等价无价小 1 3、x = 0 是函数y = (1 -sinx)紺勺() A连续点E可去间断点C跳跃间断点D无穷型间断点 4、下列数列有极限并且极限为1的选项为( ) n 1 n A X n ( 1)n B X n si n - n n 2 1 1 C X n-(a 1) D X n cos a n 5、若f "(x)在X0处取得最大值，则必有() A f /(X。)o Bf /(X。)o Cf /(X。)0且f''( X o) 1 In x 1 ; 2 y x 3 2x 2 x ;3 y 也厂,?1)^ 4?0) lim (x 1)(x m) 5 解：原式=x 1 (x 1)(x 3) m 7 b limU 」2 x 1 x 3 4 7,a 6 1、 2、 、判断题 无穷多个无穷小的和是无穷小 lim 沁在区间(， X 0 X 是连续函数() 3、 f"(x 0)=0—定为f(x)的拐点 () 4、 若f(X)在X o 处取得极值，则必有 f(x)在X 0处连续不可导( ) 5、 (x) 在 0,1 f '(x) 0令 A f'(0) f'(1),C f(1) f (0),则必有 A>B>C() 1~5 FFFFT 二、计算题 1用洛必达法则求极限 x im 0 1 e x2 解：原式=lim x 0 1 x lim e x2 ( 2x x 0 J 2x 3 1 lim e x x 0 2 若 f (x) (x 3 10)4,求f ''(0) 解: 4( x 3 24x f'(x) f ''(x) f ''(x) 0 3 2 2 , 3 10) 3x 12x (x .3 3 2 3 (x 10) 12x 3 (x 10) 3x 10)3 3 . 3 3 4 , 3 2 24x (x 10) 108x (x 10) 4 I o 2 3 求极限 lim(cos x) x x 0 . 《微积分》试题 一、选择题（3×5=15） 1、.函数f (x)=1＋x3＋x5,则f (x3＋x5)为( d ) (A)1＋x3＋x5(B)1＋2(x3＋x5) (C)1＋x6＋x10(D)1＋(x3＋x5)3＋(x3＋x5)5 2、.函数f(x)在区间[a,b] 上连续，则以下结论正确的是( b ) (A)f (x)可能存在，也可能不存在，x∈[a，b]。 (B)f (x)在[a，b] 上必有最大值。 (C)f (x)在[a，b] 上必有最小值，但没有最大值。 (D)f (x)在(a,b) 上必有最小值。 3、函数的弹性是函数对自变量的（ C ） A、导数 B、变化率 C、相对变化率 D、微分 4、下列论断正确的是（ a ） A、可导极值点必为驻点 B、极值点必为驻点 C 、 驻点必为可导极值点 D 、 驻点必为极值点 5、∫e －x dx=( b ) (A)e －x ＋c (B)－e －x ＋c (C)－e －x (D)－e x ＋c 二、填空题（3×5=15） 1.设，则 。 [答案： ] 2．函数ｙ＝ｘ＋ｅx 上点 (0,1) 处的切线方程是_____________。[答案：２ｘ－ｙ＋１＝０] 3、物体运动方程为S=1 1+t （米）。则在t=1秒时，物体速度为V=＿＿＿＿，加速度 为a=＿＿＿＿。[答案：41-，4 1 ] 4.设,则 。 [答案： 3 4] 5．若? +=c e 2dx )x (f 2 x ，则 f(x)=_________。[答案：2 x e ] 三、计算题 1、设x sin e y x 1tan = ，求dy 。 （10分） 解：dy=d x sin e x 1tan =dx x sin x 1sec x 1x cos e 22x 1tan ?? ? ??- 2．计算 ?+2x )e 1(dx 。 （15分） 解：原式＝?+-+dx )e 1(e e 12x x x ＝??++-+2x x x )e 1()e 1(d e 1dx ＝?+++-+x x x x e 11 dx e 1e e 1 ＝x-ln(1+e x )+x e 11 + +c 3.求 （15分） 解： 4．设一质量为ｍ的物体从高空自由落下，空气阻力正比于速度（ 比例常数为ｋ）０ ）求 速 度 与 时 间 的 关 系 。 （15分） 解：设速度为ｕ，则ｕ满足ｍ＝dt du ＝ｍｇ－ｋｕ 解方程得ｕ＝ k 1 （ｍｇ－ｃｅ-kt/m ） 由ｕ│t=0＝０定出ｃ，得ｕ＝k mg （１－ｅ-kt/m ）微积分考试试题
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