Language Teaching:
A Scheme. for Teacher Education Editors: C N Callan and H G Widdowson
Syllabus Design David Nunan
The author and series editors vi Introduction vii
Section One: Defining syllabus design1 1The scope of syllabus design3 1.1Introduction3 1.2 A general curricnlum model4 1.3Defining 'syllabus'5 1.4The role of the classroom teacher7 1.5Conclusion8 2Points of departure10 2.1Introduction10 2.2Basic orientations11 2.3Learning purpose13 2.4Learning goals24 2.5Conclusion25 3Product-oriented syllabuses27 3.1Introduction27 3.2Analytic and synthetic syllabus planning27 3.3Grammatical syllabuses28 3.4Criticizing grammatical syllabuses30 3.5Functional-notional syllabuses35 3.6Criticizing functional-notional syllabuses36 3.7Analytic syllabuses37 3.8Conclusion39 4Process-oriented syllabuses40 4.1Introduction40 4.2Procedural syllabuses42 4.3Task-based syllabuses44
4.4Content syllabuses48 4.5The natural approach51 4.6Syllabus design and methodology52 4.7Grading tasks54 4.8Conclusion60 5Objectives61
5.1Introduction61 5.2Types of objective61 5.3Performance objectives in language teaching63 5.4Criticizing performance objectives67 5.5Process and product objectives69 5.6Conclusion71
Section Two: Demonstrating syllabus design73 6Needs and goals75 6.1Introduction75 6.2Needs analysis75 6.3From needs to goals79 6.4Conclusion84 7Selecting and grading content85 7.1Introduction85 7.2Selecting grammatical components86 7.3Selecting functional and notional components87 7.4Relating grammatical, functional, and notional
components87 7.5Grading content92 7.6Conclusion95 8Selecting and grading learning tasks96 8.1Introduction96 8.2Goals, objectives, and tasks96 8.3Procedural syllabuses98 8.4The natural approach102 8.5Content-based syllabuses104 8.6Levels of difficulty107 8.7Teaching grammar as process118 8.8Conclusion121
9Selecting and grading objectives122 9.1Introduction)122 9.2Product-oriented objectives122 9.3Process-oriented objectives131 9.4Conclusion133
Section Three: Exploring syllabus design135 10General principles137 10.1Curriculum and syllabus models137 10.2Purposes and goals140 10.3Syllabus products144 10.4Experiential content147 10..5Tasks and activities149 10.6Objectives153
Glossary158
Further reading160
Bibliography161
Index166
SECTION ONE
Defining syllabus design
The scope of syllabus design
1.1 Introduction
We will start by outlining the scope of syllabus design and relating it to the broader field of curriculum development. Later, in 1.4, we shall also look at the role of the teacher in syllabus design.
Within the literature, there is some confusion over the terms 'syllabus' and 'curriculum'. It would, therefore, be as well to give some indication at the outset of what is meant here by syllabus, and also how syllabus design is related to curriculum development.
? TASK 1
As a preliminary activity, write a short definition of the terms
'syllabus' and 'curriculum'.
In language teaching, there has been a comparative neglect of systematic curriculum development. In particular, there have been few attempts to apply, in any systematic fashion, principles of curriculum development to the planning, implementation, and evaluation of language programmes.
Language curriculum specialists have tended to focus on only part of the total picture — some specializing in syllabus design, others in methodolo n
fragmented approach has been criticized, and there have been calls for a more comprehensive approach to language curriculum design (see, for example, Breen and Candlin 1980; Richards 1984; Nunan 1985). The present book is intended to provide teachers with the skills they need to address, in a systematic fashion, the problems and tasks which confront them in their programme planning.
Candl in (1984) suggests that curricula are concerned with making general statements about language learning, learning purpose and experience, evaluation, and the role relationships of teachers and learners. According to Candlin, they will also contain banks of learning items and suggestions about how these might be used in class. Syllabuses, on the other hand, are more localized and are based on accounts and records of what actually happens at die classroom level as teachers and learners apply a given curriculum to their own situation. These accounts can be used to make subsequent modifications to the curriculum, so that the developmental process is ongoing and cyclical.
1.2 A general curriculum model
TASK 2
Examine the following planning tasks and decide on the order in
which they might be carried out.
—monitoring and assessing student progress
— selecting suitable materials
—stating the objectives of the course
—evaluating the course
—listing grammatical and functional components
—designing learning activities and tasks
—instructing students
—identifying topics, themes, and situations
h is possible to study 'the curriculum' of an educational institution from a
number of different perspectives. lo the first instance we can look at curriculum planning, that is at decision making, in relation to identifying learners' needs and purposes; establishing goals and objectives; selecting and grading content; organizing appropriate learning arrangements and learner groupings; selecting, adapting, or developing appropriate mate-
rials, learning tasks, and assessment and evaluation tools.
Alternatively, we can study the curriculum 'in action' as it were. This second perspective takes us into the classroom itself. Here we can observe the teaching/learning process and study the ways in which the intentions of the curriculum planners, which were developed during the planning phase, are translated into action.
Yet another perspective relates to assessment and evaluation. From this perspective, we would try and find out what students had learned and what they had failed to learn in relation us what had been planned. Additionally, we might want to find out whether they had learned anything which had not been planned. We would also want to account for our findings, to make judgements about why some things had succeeded and others had failed, and perhaps to make recommendations about what changes might be made to improve things in the future.
Filially, we might want to study the management of the teaching institution, looking at the resources available and how these are utilized, hose the institution relates to and responds to the wider community, how constraints imposed by limited resources and the decisions of administra-tors affect what happens in the classroom, and so on.
All of these perspectives taken together represent the field of curriculum study. As we can see, the field is a large .d complex one.
It is important that, in the planning, implementation, and evaluation of a given curriculum, all elements be integrated, so that decisions made at one
level are not in conflict with those made at another. For instance, in courses based on principles of communicative language teaching, it is important that these principles are reflected, not only in curriculum documents and syllabus plans, but also in classroom activities, patterns of classroom interaction, and in tests of communicative performance.
1.3 Defining 'syllabus'
There are several conflicting views on just what it is that distinguishes syllabus design from curriculum development. There is also some disagreement about the nature of 'the syllabus'. In books and papers on the subject, it is possible to distinguish a broad and a narrow approach to syllabus design.
The narrow view draws a clear distinction between syllabus design and methodology. Syllabus design is seen as being concerned essentially with the selection and grading of content, while methodology is concerned with the selection of learning tasks and activities. Those who adopt a broader communicative language teaching the distinction between content and tasks is difficult to sustain.
The following quotes have been taken from Brumfit (1984) which provides an excellent overview of the range and diversity of opinion on syllabus design. The broad and narrow views are both represented in the book, as you will see from the quotes.
■ TASK 3
As you read the quotes, see whether you can identify which writers
are advocating a broad approach and which a narrow approach.
1 . . .I would like to draw attention to a distinction . . . between
curriculum or syllabus, that is its content, structure, parts and
organisation, and, . . t.vhat in curriculum theory is often called
curriculum processes, that is curriculum development, imple-
mentation, dissemination and evaluation. The former is con-
cerned with the WHAT of curriculum: what the curriculum is like
or should be like; the latter is concerned with the WHO and
HOW of establishing the curriculum.
(Stern 1984: 10-11)
2 [The syllabus] replaces the concept of 'method', and the syllabus
is now seen as an instrument by which the teacher, with the help
of the syllabus designer, can achieve a degree of 'fit' between the
needs and aims of the learner (as social being and as individual)
and the activities which will take place in the classroom.
(Yalden 1984: 14)
3 ... the syllabus is simply a framework within which activities can
be carried mit: a teaching device to facilitate learning. It only
becomes a threat to pedagogy when it is regarded as absolute rules
for determining what is to be learned rather than points of
reference from which bearings can be taken.
(Widclowson 1984: 26)
4 We might . .. ask whether it is possible to separate so easily what
we have been calling content from what we have been calling
method or procedure, or indeed whether we can avoid bringing
evaluation into the debate?
5 Any syllabus will express—however indirectly—certain assump-
tions about language, about the psychological process of learn-
ing, and about the pedagogic and social processes within a
classroom.
(Breen 1984: 49)
6 . . . curriculum is a very general concept which involves
consideration of the whole complex of philosophical, social and
administrative factors which contribute to the planning of an
educational program. Syllabus, on the other hand, refers to that
subpart of curriculum which is concerned with a specification of
what units will be taught (as distinct from how they will be
taught, which is a matter for methodology).
(Allen 1984: 61)
7 Since language is highly complex and cannot be taught all at the
same time, successful teaching requires that there should be a
selection of material depending on the prior definition of
objectives, proficiency level, and duration of course. This
selection takes place at the syllabus planning stage.
(op. cit.: 65)
As you can see, some language specialists believe that syllabus (the selection and grading of content) and methodology should be kept separate; others think otherwise. One of the issues you will have to decide on as you work through this book is whether you thi iik syllabuses should be defined solely in terms of the selection and grading of contera, or whether they should also attempt to specify and grade learning tasks and activities.
Here, we shall take as our point of departure the rather traditional notion that a syllabus is a statement of content which is used as the basis for planning courses of various kinds, and that the task of the syllabus designer is to select and grade this content. To begin with, then, we shall distinguish between syllabus design, which is concerned with the 'what of a language
programme, and methodology, which is concerned with the 'how'. (Later, we shall look at proposals for 'procedural' syllabuses in which the distinction between the 'what' and the 'how' becomes difficult to sustain.) One document which gives a detailed account of the various syllabus components which need to be corisidered in developing language courses is Threshold Level English (van Ek 1975). van Ek lists the following as necessary components of a language syllabus:
1 the situations in which the foreign language will be used, includ-
ing the topics which trill be dealt with;
2 the language activities in which the learner will engage;
3 the language functions which the learner will fulfil;
4 what the learner will be able to do with respect to each topic;
5 the general notions which the learner will be able to handle;
6 the specific (topic-related) notions which the learner will be able
to handle;
7 the language forms which the learner will be able to use;
8 the degree of skill with which the learner will be able to perform.
(van Ek 1975: 8-9)
?TASK 4
Do you think that van Ek subscribes to a 'broad' or 'narrow' view of
syllabus design?
Which, if any, of the above components do you think are beyond the
scope of syllabus design?
1.4 The role of the classroom teacher
In a recent book dealing, among other things, with syllabus design issues, Bell (1983) claims that teachers are, in the main, consumers of other people's syllabuses; in other words, that their role is to implement the plans of applied linguists, government agencies, and so on. While some teachers have a relatively free hand in designing the syllabuses on which their teaching programmes are based, most are likely to be, as Bell suggests, consumers of other people's syllabuses.
?
Study the following list of planning tasks.
In your experience, for which of these tasks do you see the classroom
teacher as having primary responsibility?
Rate each task on a scale from 0 (no responsibility) to 5 (total
responsibility).
—identifying learners' communicative needs0 1 2 3 4 5
—selecting and grading syllabus content0 1 2 3 4 5
— grouping learners into different classes
or learning arrangements0 1 2 3 4 5— selecting/creating materials and learning
activities0 1 2 3 4 5—monitoring and assessing learner progress0 1 2 3 4 5
—course evaluation0 1 2 3 4 5 In a recent study of an educational system where classroom teachers are expected to design, implement, and evaluate their own curriculum, one group of teachers, when asked the above question, stated that they saw themselves as having primary responsibility for all of the above tasks except for the third one (grouping learners). Some of the teachers in the system felt quite comfortable with an expanded professional role. Others felt that syllabus development should be carried out by people with specific expertise, and believed that they were being asked to undertake tasks for which they were not adequately trained (Nunan 1987).
■ TASK 6
What might be the advantages andlor disadvantages of teachers in
your system designing their own syllabuses?
Can you think of any reasons why teachers might be discouraged
from designing, or might not want to design their own syllabuses?
Are these reasons principally pedagogic, political, or administra 1.5 Conclusion
In 1, I have tried to provide some idea of the scope of syllabus design. I have suggested that traditionally syllabus design has been seen as a subsidiary component of curriculum design. 'Curriculum' is concerned with the planning, implementation, evaluation, management, and administration of education programmes. 'Syllabus', on the other hand, focuses more narrowly on the selection and grading of content.
While it is realized that few teachers are in the position of being able to design their own syllabuses, it is hoped that most are in a position to interpret and modify their syllabuses in the process of translating them into action. The purpose of this book is therefore to present the central issues and options available for syllabus design in order to provide teachers with the necessary knowledge and skills for evaluating, and, where feasible, modifying and adapting the syllabuses with which they work. At the very least, this book should help you understand (and therefore more effectively exploit) the syllabuses and course materials on which your programmes are based.
■ TASK 7
Look back at the definitions you wrote in Task 1 and rewrite these in the light of the information presented in 1.
In what ways, if any, do your revised definitions differ from the ones
you wrote at the beginning?
In 2, we shall look at some of the starting points in syllabus design. The next central question to be addressed is, 'Where does syllabus content come from?' In seeking answers to this question, we shall look at techniques for obtaining information from and about learners for use in syllabus design. We shall examine the controversy which exists over the very nature of language itself ancl how this influences the making of decisions about what to include in the syllabus. We shall also look at the distinction between product-oriented and process-oriented approaches to syllabus design. These two orientations are studied in detail in 3 and 4. The final part of Section One draws on the content of the preceding parts and relates this content to the issue of objectives. You will be asked to consider whether or not we need objectives, and if so, how these should be formulated.
2 Points of departure
2.1 Introduction
In 1 it was argued that syllabus design was essentially concerned with the selection and grading of content. As such, it formed a sub-component of the planning phase of curriculum development. (You will recall that the curriculum has at least three phases: a planning phase, an implementation phase, and an evaluation phase.)
The first question to confront the syllabus designer is where the content is to come from in the first place. We shall now look at the options available to syllabus designers in their search for starting points in syllabus design.
■ TASK 8
Can you think of any ways in which our beliefs about the nature of
language and learning might in fl uence our decision-making on what
to put into the syllabus and how to grade it?
If we had consensus on just what it was that we were supposed to teach in order for learners to develop proficiency in a second or foreign language; if we knew a great deal more than we do about language learning; if it were possible to teach the totality of a given language, and if we had complete descriptions of the target language, problems associated with selecting and sequencing content and learning experiences would be relatively straight-forward. As it happens, there is not a great deal of agreement within the teaching profession on the nature of language and language learning. As a consequence, we must make judgements in selecting syllabus components from all the options which are available to us. As Breen (1984) points out, these judgements are not yalue-free, but reflect our beliefs about the nature of language and learning. In this and the other parts in this section, we shall see how value judgements affect decision-making in syllabus design.
The need to make value judgements and choices in deciding what to include in (or omit from) specifications of content and which elements are to be the basic building blocks of the syllabus, presents syllabus designers with constant problems. The issue of content selection becomes particularly pressing if the syllabus is intended to underpin short courses. (It could be argued that the shorter the course, the greater the need for precision in content specification.)
2.2 Basic orientations
Until fairly recently, most syllabus designers started out by drawing up lists of grammatical, phonological, and vocabulary items which were then graded according to difficulty and usefulness. The task for the learner was seen as gaining mastery over these grammatical, phonological, and vocabulary items.
Learning a language, it was assumed, entails mastering the elements
or building blocks of the language and learning the rules by whirls
these elements are combined, from phoneme to morpheme to word
to phrase to sentence.
(Richards and Rodgers 1986: 49)
During the 1970s, communicative views of language teaching began to be incorporated into syllabus design. The central cpiestion for proponents of this new view was, 'What does the learner wantineed to do with the target language?' rather than, 'What are the linguistic elements which the learner needs to master?' Syllabuses began to appear in whirls content was specified, not only in terms of the grammatical elements whirls the learners were expected to master, but also in terms of the functional skills they would need to master in order to communicate successfully.
This movement led in part to the development of English for Specific Purposes (ESP). Here, syllabus designers focused, not only on language functions, but also on experiential content (that is, the subject matter through which the language is taught).
Traditionally, linguistically-oriented syllabuses, along with many so-called communicative syllabuses, shared one thing in common: they tended to focus on the things that learners should know or be able to do as a result of instruction. In the rest of this book we shall refer to syllabuses in which content is stated in terms of the outcomes of instruction as 'product-oriented'.
As we have already seen, a distinction is traditionally drawn between syllabus design, which is concerned with outcomes, and methodology, which is concerned with the process through which these outcomes are to be brought about. Recently, however, some syllabus designers have suggested that syllabus content might be specified in terms of learning tasks and activities. They justify this suggestion on the grounds that communica-tion is a process rather than a set of products.
In evaluating syllabus proposals, we have to decide whether this view represents a fundamental change in perspective, or whether those advocating process syllabuses have made a category error; whether, in fact, they are really addressing methodological rather than syllabus issues. This is something which you will have to decide for yourself as you work through this book.
■ TASK 9
At this stage, what is your view on the legitimacy of defining syllabuses in terms of learning processes? Do you think that syllabuses should list and grade learning tasks and activities as well as linguistic content?
A given syllabus will specify all or some of the following: grammatical structures, functions, notions, topics, themes, situations, activities, and tasks. Each of these elements is either product or process oriented, and the inclusion of each will be justified according to beliefs about the nature of language, the needs of the learner, or the nature of learning.
In the rest of this book, we shall be making constant references to and comparisons between process and product. What we mean when we refer to 'process' is a series of actions directed toward some end. The 'product' is the end itself. This may be clearer if we consider some examples. A list of grammatical structures is a product. Classroom drilling undertaken by learners in order to learn the structures is a process. The interaction of two speakers as they communicate with each other is a process. A tape recording of their conversation is a product.
Did you find that some elements could be assigned to more than one orientation or point of reference? Which were these?
2.3 Learning purpose
In recent years, a major trend in language syllabus design has been the use of information from and about learners in curriculum decision-making. In this section, we shall look at some of the ways in which learner data have been used to inform decision-making in syllabus design. In the course of the discussion we shall look at the controversy over general and specific purpose syllabus design.
Assumptions about the learner's purpose in undertaking a language cosiese, as well as the syllabus designer's beliefs about the nature of language and learning can have a marked influence on the shape of the syllabus on which the course is based. Learners' purposes will vary according to how specific they are, and how immediately learners wish to employ their developing language skills.
■ TASK 11
Which of the following statements represent specific language needs
and which are more general?
'I want to be able to talk to my neighbours in English.'
'I want to study microbiology in an English-speaking university.'
'I want to develop an appreciation of German culture by studying
the language.'
'I want to be able to communicate in Greek.'
'I want my daughter to study French at school so she can matriculate
and read French at university.'
'I want to read newspapers in Indonesian.'
'I want to understand Thai radio broadcasts.'
'I need "survival" English.'
'I want to be able to read and appreciate French literature.'
'I want to get a better job at the factory.'
'I want to speak English.'
'I want to learn English for nursing.'
For which of the above would it be relatively easy to predict the
grammar and topics to be listed in the syllabus?
For which would it be difficult to predict the grammar and topics?
Techniques and procedures for collecting information to be used in syllabus design are referred to as needs analysis. The se techniques have been borrowed and adapted from other areas of training and development, particularly those assoeiated with industry and technology.
? TASK 12
One general weakness of most of the literature on needs analysis is the tendency to think only in terms of learner needs. Can you think of any other groups whose needs should be considered? Information will need to be collected, not only on why learners want to learn the target language, but also about such things as societal expectations and constraints and the resources available for implementing the syllabus.
Broadly speaking, there arc two different types of needs analysis used by language syllabus designers. The first of these is learner analysis, while the second is task analysis.
Learner analysis is based on information about the learner. The central question of concern to the syllabus designer is: Tor what purpose or purposes is the learner learning the language?' There are many other subsidiary questions, indeed it is possible to collect a wide range of information as can be seen from the following data collection forms.
1* TASK 13
Which of the above information do you think is likely to be most useful for planning purposes?
What are some of the purposes to which the information might be put?
The information can serve many purposes, depending on the nature of the educational institution in which it is to be used. In the first instance, it can guide the selection of content. It may also be used to assign learners to class
groupings. This will be quite a straightforward matter if classes are based
大一高等数学期末考试卷（精编试题）及答案详解 一、单项选择题 (本大题有4小题, 每小题4分, 共16分) 1. )( 0),sin (cos )( 处有则在设=+=x x x x x f . （A ）(0)2f '= （B ）(0)1f '=（C ）(0)0f '= （D ）()f x 不可导. 2. ）时（ ，则当，设133)(11)(3→-=+-= x x x x x x βα. （A ）()()x x αβ与是同阶无穷小，但不是等价无穷小； （B ）()()x x αβ与是 等价无穷小； （C ）()x α是比()x β高阶的无穷小； （D ）()x β是比()x α高阶的无穷小. 3. 若 ()()()0 2x F x t x f t dt =-?，其中()f x 在区间上(1,1)-二阶可导且 '>()0f x ，则（ ）. （A ）函数()F x 必在0x =处取得极大值； （B ）函数()F x 必在0x =处取得极小值； （C ）函数()F x 在0x =处没有极值，但点(0,(0))F 为曲线()y F x =的拐点； （D ）函数()F x 在0x =处没有极值，点(0,(0))F 也不是曲线()y F x =的拐点。 4. ) ( )( , )(2)( )(1 =+=?x f dt t f x x f x f 则是连续函数，且设 （A ）2 2x （B ）2 2 2x +（C ）1x - （D ）2x +. 二、填空题（本大题有4小题，每小题4分，共16分） 5. = +→x x x sin 20 ) 31(lim . 6. ,)(cos 的一个原函数是已知 x f x x =? ?x x x x f d cos )(则 . 7. lim (cos cos cos )→∞ -+++=2 2 2 21 n n n n n n π π ππ . 8. = -+? 2 12 12 211 arcsin － dx x x x . 三、解答题（本大题有5小题，每小题8分，共40分） 9. 设函数=()y y x 由方程 sin()1x y e xy ++=确定，求'()y x 以及'(0)y . 10. .d )1(17 7 x x x x ?+-求
大一高等数学期末考试试卷 一、选择题（共12分） 1. （3分）若2,0,(),0 x e x f x a x x ?<=?+>?为连续函数,则a 的值为( ). (A)1 (B)2 (C)3 (D)-1 2. （3分）已知(3)2,f '=则0(3)(3)lim 2h f h f h →--的值为（ ）. (A)1 (B)3 (C)-1 (D) 12 3. （3 分）定积分22 ππ-?的值为（ ）. (A)0 (B)-2 (C)1 (D)2 4. （3分）若()f x 在0x x =处不连续,则()f x 在该点处( ). (A)必不可导 (B)一定可导(C)可能可导 (D)必无极限 二、填空题（共12分） 1．（3分） 平面上过点(0,1)，且在任意一点(,)x y 处的切线斜率为23x 的曲线方程为 . 2. （3分） 1 241(sin )x x x dx -+=? . 3. （3分） 201lim sin x x x →= . 4. （3分） 3223y x x =-的极大值为 . 三、计算题（共42分） 1. （6分）求2 0ln(15)lim .sin 3x x x x →+ 2. （6 分）设2,1 y x =+求.y ' 3. （6分）求不定积分2ln(1).x x dx +? 4. （6分）求3 0(1),f x dx -?其中,1,()1cos 1, 1.x x x f x x e x ?≤?=+??+>?
5. （6分）设函数()y f x =由方程00cos 0y x t e dt tdt +=??所确定,求.dy 6. （6分）设2()sin ,f x dx x C =+?求(23).f x dx +? 7. （6分）求极限3lim 1.2n n n →∞??+ ??? 四、解答题（共28分） 1. （7分）设(ln )1,f x x '=+且(0)1,f =求().f x 2. （7分）求由曲线cos 2 2y x x ππ??=-≤≤ ???与x 轴所围成图形绕着x 轴旋转一周所得旋转体的体积. 3. （7分）求曲线3232419y x x x =-+-在拐点处的切线方程. 4. （7 分）求函数y x =+[5,1]-上的最小值和最大值. 五、证明题(6分) 设()f x ''在区间[,]a b 上连续,证明 1()[()()]()()().22b b a a b a f x dx f a f b x a x b f x dx -''=++--?? 标准答案 一、 1 B; 2 C; 3 D; 4 A. 二、 1 31;y x =+ 2 2;3 3 0; 4 0. 三、 1 解 原式2 05lim 3x x x x →?= 5分 53 = 1分 2 解 22ln ln ln(1),12 x y x x ==-++Q 2分 2212[]121 x y x x '∴=-++ 4分
大一高等数学期末考试试卷 （一） 一、选择题（共12分） 1. （3分）若2,0, (),0x e x f x a x x ?<=?+>? 为连续函数,则a 的值为( ). (A)1 (B)2 (C)3 (D)-1 2. （3分）已知(3)2,f '=则0 (3)(3) lim 2h f h f h →--的值为（ ）. (A)1 (B)3 (C)-1 (D) 12 3. （3分）定积分 22 π π - ?的值为（ ）. (A)0 (B)-2 (C)1 (D)2 4. （3分）若()f x 在0x x =处不连续,则()f x 在该点处( ). (A)必不可导 (B)一定可导(C)可能可导 (D)必无极限 二、填空题（共12分） 1．（3分） 平面上过点(0,1)，且在任意一点(,)x y 处的切线斜率为2 3x 的曲线方程为 . 2. （3分） 1 241 (sin )x x x dx -+=? . 3. （3分） 2 1 lim sin x x x →= . 4. （3分） 3 2 23y x x =-的极大值为 . 三、计算题（共42分） 1. （6分）求2 ln(15) lim .sin 3x x x x →+ 2. （6分）设y =求.y ' 3. （6分）求不定积分2 ln(1).x x dx +?
4. （6分）求 3 (1),f x dx -? 其中,1,()1cos 1, 1.x x x f x x e x ?≤? =+??+>? 5. （6分）设函数()y f x =由方程0 cos 0y x t e dt tdt +=? ?所确定,求.dy 6. （6分）设 2 ()sin ,f x dx x C =+?求(23).f x dx +? 7. （6分）求极限3lim 1.2n n n →∞? ?+ ??? 四、解答题（共28分） 1. （7分）设(ln )1,f x x '=+且(0)1,f =求().f x 2. （7分）求由曲线cos 2 2y x x π π??=- ≤≤ ???与x 轴所围成图形绕着x 轴旋转一周所得旋 转体的体积. 3. （7分）求曲线32 32419y x x x =-+-在拐点处的切线方程. 4. （7 分）求函数y x =+[5,1]-上的最小值和最大值. 五、证明题(6分) 设()f x ''在区间[,]a b 上连续,证明 1()[()()]()()().22b b a a b a f x dx f a f b x a x b f x dx -''=++--? ? （二） 一、 填空题（每小题3分，共18分） 1．设函数()2 31 22+--=x x x x f ，则1=x 是()x f 的第 类间断点. 2．函数( )2 1ln x y +=，则='y . 3． =? ? ? ??+∞→x x x x 21lim . 4．曲线x y 1=在点?? ? ??2,21处的切线方程为 .
( 大一上学期高数期末考试 一、单项选择题 (本大题有4小题, 每小题4分, 共16分) 1. )( 0),sin (cos )( 处有则在设=+=x x x x x f . （A ）(0)2f '= （B ）(0)1f '=（C ）(0)0f '= （D ）()f x 不可导. 2. ） 时（ ，则当，设133)(11)(3→-=+-=x x x x x x βα. （A ）()()x x αβ与是同阶无穷小，但不是等价无穷小； （B ）()()x x αβ与是 等价无穷小； （C ）()x α是比()x β高阶的无穷小； （D ）()x β是比()x α高阶的无穷小. 3. … 4. 若 ()()()0 2x F x t x f t dt =-?，其中()f x 在区间上(1,1)-二阶可导且 '>()0f x ，则（ ）. （A ）函数()F x 必在0x =处取得极大值； （B ）函数()F x 必在0x =处取得极小值； （C ）函数()F x 在0x =处没有极值，但点(0,(0))F 为曲线()y F x =的拐点； （D ）函数()F x 在0x =处没有极值，点(0,(0))F 也不是曲线()y F x =的拐点。 5. ) ( )( , )(2)( )(1 =+=?x f dt t f x x f x f 则是连续函数，且设 （A ）22x （B ）2 2 2x +（C ）1x - （D ）2x +. 二、填空题（本大题有4小题，每小题4分，共16分） 6. , 7. = +→x x x sin 20 ) 31(lim . 8. ,)(cos 的一个原函数是已知 x f x x =? ?x x x x f d cos )(则 . 9. lim (cos cos cos )→∞ -+++=2 2 2 21 n n n n n n π π ππ . 10. = -+? 2 12 1 2 211 arcsin － dx x x x . 三、解答题（本大题有5小题，每小题8分，共40分） 11. 设函数=()y y x 由方程 sin()1x y e xy ++=确定，求'()y x 以及'(0)y .
大一上学期高数期末考试 一、单项选择题 (本大题有4小题, 每小题4分, 共16分) 1. )( 0),sin (cos )( 处有则在设=+=x x x x x f . （A ）(0)2f '= （B ）(0)1f '=（C ）(0)0f '= （D ）()f x 不可导. 2. ） 时（ ，则当，设133)(11)(3→-=+-=x x x x x x βα. （A ）()()x x αβ与是同阶无穷小，但不是等价无穷小； （B ）()() x x αβ与是等价无穷小； （C ）()x α是比()x β高阶的无穷小； （D ）()x β是比()x α高阶的无穷小. 3. 若 ()()()0 2x F x t x f t dt =-?，其中()f x 在区间上(1,1)-二阶可导且 '>()0f x ，则（ ）. （A ）函数()F x 必在0x =处取得极大值； （B ）函数()F x 必在0x =处取得极小值； （C ）函数()F x 在0x =处没有极值，但点(0,(0))F 为曲线()y F x =的拐点； （D ）函数()F x 在0x =处没有极值，点(0,(0))F 也不是曲线()y F x =的拐点。 4. ) ( )( , )(2)( )(1 =+=?x f dt t f x x f x f 则是连续函数，且设 （A ）22x （B ）2 2 2x +（C ）1x - （D ）2x +. 二、填空题（本大题有4小题，每小题4分，共16分） 5. = +→x x x sin 2 ) 31(lim . 6. ,)(cos 的一个原函数是已知 x f x x =? ?x x x x f d cos )(则 . 7. lim (cos cos cos )→∞-+++= 2 2 221 n n n n n n ππ ππ . 8. = -+? 2 12 12 211 arcsin － dx x x x . 三、解答题（本大题有5小题，每小题8分，共40分） 9. 设函数=()y y x 由方程 sin()1x y e xy ++=确定，求'()y x 以及'(0)y . 10. .d )1(17 7 x x x x ?+-求
高等数学II 填空题 1、()1 3 1sin x x dx -+=? _______________________. 2、设()1 1 x x f x e dx e C =+?, 则()f x =_________________. 3、微分方程2220d y dy y dx dx -+=的通解为_______________________. 4、函数 (,)ln 1f x y x y =--_______________. 5、椭圆22 1169 x y += 绕x 轴旋转一周所得旋转体的体积为______________________. 计算题 1、计算不定积分 2211sec dx x x ?. 2 、计算不定积分 dx , ()0a >. 3、计算定积分 320sin cos x x dx π? 4、计算定积分 1 0arcsin x dx ? 解答题 1、设函数()f x 的原函数()F x 恒正, (0)1F =且()()f x F x x =, 且()f x 的表达式. 2、解微分方程()52211dy y x dx x =+++，并求出其满足初始条件01|3 x y ==-的特解. 3、设2ln z u v =，且x u y =， 32v x y =-, 求z x ??和z y ??, 并写出dz . 4、设02 (), 0() , 0 x tf t dt x F x x A x ??≠=??=??, 其中()f x 具有连续导数且(0)0f =. (1) 如果()F x 在点0x =处连续, 求A 的值; (2) 在(1)的前提下, 证明()F x 在点0x =处可导, 并求(0)F '的值.
（2010至2011学年第一学期） 课程名称： 高等数学(上)(A 卷) 考试(考查)： 考试 2008年 1 月 10日 共 6 页 1、 满分100分。要求卷面整洁、字迹工整、无错别字。 2、 考生必须将姓名、班级、学号完整、准确、清楚地填写在试卷规定的地方，否 则视为废卷。 3、 考生必须在签到单上签到，若出现遗漏，后果自负。 4、 如有答题纸，答案请全部写在答题纸上，否则不给分；考完请将试卷和答题卷 分别一同交回，否则不给分。 试 题 一、单选题（请将正确的答案填在对应括号内，每题3分，共15分） 1. =--→1 ) 1sin(lim 21x x x ( ) (A) 1； (B) 0； (C) 2； (D) 2 1 2.若)(x f 的一个原函数为)(x F ，则dx e f e x x )(? --为( ) (A) c e F x +)(； (B) c e F x +--)(； (C) c e F x +-)(； (D ) c x e F x +-) ( 3.下列广义积分中 ( )是收敛的. (A) ? +∞ ∞ -xdx sin ； (B)dx x ? -1 11； (C) dx x x ?+∞∞-+2 1； (D)?∞-0dx e x 。 4. )(x f 为定义在[]b a ,上的函数，则下列结论错误的是( ) (A) )(x f 可导，则)(x f 一定连续； (B) )(x f 可微，则)(x f 不一定
可导； (C) )(x f 可积（常义），则)(x f 一定有界； (D) 函数)(x f 连续，则? x a dt t f )(在[]b a ,上一定可导。 5. 设函数=)(x f n n x x 211lim ++∞→ ，则下列结论正确的为( ) (A) 不存在间断点； (B) 存在间断点1=x ； (C) 存在间断点0=x ； (D) 存在间断点1-=x 二、填空题（请将正确的结果填在横线上.每题3分，共18分） 1. 极限=-+→x x x 1 1lim 20 _____. 2. 曲线???=+=3 2 1t y t x 在2=t 处的切线方程为______. 3. 已知方程x xe y y y 265=+'-''的一个特解为x e x x 22 )2(2 1+- ，则该方程的通解为 . 4. 设)(x f 在2=x 处连续，且22 ) (lim 2=-→x x f x ，则_____)2(='f 5．由实验知道，弹簧在拉伸过程中需要的力F （牛顿）与伸长量s 成正比，即ks F =（k 为比例系数），当把弹簧由原长拉伸6cm 时，所作的功为_________焦耳。 6．曲线23 3 2 x y =上相应于x 从3到8的一段弧长为 . 三、设0→x 时，)(22 c bx ax e x ++-是比2 x 高阶的无穷小，求常数c b a ,,的值（6分）
2019-2020 学年第二学期试卷（A 卷） 课程：《高等数学》 一、填空题：（每空 3分，共 30分） （说明：将运算结果．．．． 填写在每小题相应的横线） 1．设函数22 ()30 x x f x x b x ?+<=? +≥? 在0x =处连续，则常数b = ． 2．如果0sin 3lim 1x x kx →=，则k = ． 3．如果()f x 在0x 处可导，则00(2)() lim x h f x h f x h →+-= ． 4．设函数1 y x = ，当x 时此函数为无穷小量，当x 时此函数为无穷大量． 5．曲线2 2 4x xy y ++= 在点(2,2)-处的切线方程为 ． 6．函数1 ()lg(5) f x x = -定义域为 ． 7．曲线3 352y x x =-++的拐点是 ． 8．曲线1 2 x y x += -的水平渐近线为 ，铅直渐近线为 ． 9．设x e -是()f x 的一个原函数，则()f x dx =? ． 10． 1 31 5sin xdx -=? ． 二、选择题：（每题5分，共 15 分） （说明：将认为正确答案的字母填写在每小题相应的括号内） 1．下列函数在1x =-处连续，但不可导的是【 】． A.1y x =+ B.2ln(1)y x =+ C. 1 1 y x = + D. 2(1)y x =+ 2．设2 11x y x -=+，则1x =-是函数的【 】． A.连续点 B. 可去间断点 C.跳跃间断点 D. 无穷间断点 3．下列等式不正确是【 】． A. 1 2 lim(12)x x x e →+= B. 110 lim(1) x x x e --→-= C. sin lim 0x x x →∞= D. 0tan lim 1x x x →=
1、(本小题5分) 求极限 lim x x x x x x →-+-+-233 21216 29124 2、(本小题5分) .d )1(2 2x x x ? +求 3、(本小题5分) 求极限limarctan 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 21 6、(本小题5分) ??. d csc cot 46x x x 求
（第七题删掉了） 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 (),2 2 9、(本小题5分) ． 求dx x x ?+3 1 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 +20 2sin 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 Λ 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) 1、(本小题7分) ,,512沿一边可用原来的石条围平方米的矩形的晒谷场某农场需建一个面积为.,,才能使材料最省多少时问晒谷场的长和宽各为另三边需砌新石条围沿 2、(本小题7分) . 823 2体积轴旋转所得的旋转体的所围成的平面图形绕和求由曲线ox x y x y == 三、解答下列各题 ( 本 大 题6分 ) 设证明有且仅有三个实根f x x x x x f x ()()()(),().=---'=1230 (答案) 一、解答下列各题 (本大题共16小题，总计77分) 1、(本小题3分)
解 大一高等数学期末考试试卷 (一)一、选择题(共12分) x,2,0,ex,fx(),1. (3分)若为连续函数,则的值为( ). a,axx，,,0,(A)1 (B)2 (C)3 (D)-1fhf(3)(3),,,2. (3分)已知则的值为( ). limf(3)2,,h,02h 1(A)1 (B)3 (C)-1 (D) 2 ,223. (3分)定积分的值为( ). 1cos,xdx,,,2 (A)0 (B)-2 (C)1 (D)2 4. (3分)若在处不连续,则在该点处( ).xx,fx()fx()0(A)必不可导(B)一定可导(C)可能可导(D)必无极限二、填空题(共12分)23x1((3分)平面上过点，且在任意一点处的切线斜率为的曲线方程(0,1)(,)xy为. 124(sin)xxxdx，,2. (3分) . ,,1 12xlimsin3. (3分) = . x,0x 324. (3分)的极大值为. yxx,,23 三、计算题(共42分) xxln(15)，lim.1. (6分)求2x,0sin3x xe,y,,2. (6分)设求y. 2x，1 2xxdxln(1).，3. (6分)求不定积分, x,3,1,x,,fxdx(1),,4. (6分)求其中()fx,1cos，x,,0x,1,1.ex，,,1 yxt5. (6分)设函数由方程所确定,求edttdt，,cos0yfx,()dy.,,0026. (6分)设求fxdxxC()sin,,，fxdx(23).，,, n3,,7. (6分)求极限lim1.，,,,,nn2,, 四、解答题(共28分)
,1. (7分)设且求fxx(ln)1,,，f(0)1,,fx(). ,,,,2. (7分)求由曲线与轴所围成图形绕着轴旋转一周所得旋 xxyxxcos,,,,,,22,, 转体的体积. 323. (7分)求曲线在拐点处的切线方程. yxxx,,，,32419 4. (7分)求函数在上的最小值和最大值. [5,1],yxx,，,1 五、证明题(6分) ,,设在区间上连续,证明fx()[,]ab bbba,1,, fxdxfafbxaxbfxdx()[()()]()()().,，，,,,,aa22 (二) 一、填空题(每小题3分，共18分) 2x,1x,1，，fx,，，1(设函数，则是的第类间断点. fx2x,3x， 则. y,y,ln1，x x2 x，1,,( 3 . ,lim,,x,, x,, 11,,y,4(曲线在点处的切线方程为. ,2,,x2,, 32,,,1,45(函数在上的最大值，最小值. y,2x,3x xarctandx,6(. ,21，x 222,，，2(函数，二、单项选择题(每小题4分，共20分) 1(数列有界是它收敛的( ) . ,,xn必要但非充分条件;充分但非必要条件;，，，，A B 充分必要条件;无关条件.，，，，C D 2(下列各式正确的是( ) .1,x,xxdx,，C; ;ln，，edx,e，C，，A B ,,x
高等数学（上）模拟试卷一 一、 填空题（每空3分，共42分） 1 、函数lg(1)y x = -的定义域是 ； 2、设函数 20() 0x x f x a x x ?<=? +≥?在点0x =连续，则a = ； 3、曲线 4 5y x =-在（-1，-4）处的切线方程是 ； 4、已知 3()f x dx x C =+? ，则()f x = ； 5、2 1lim(1) x x x →∞ -= ； 6、函数32 ()1f x x x =-+的极大点是 ； 7、设()(1)(2)2006)f x x x x x =---……(，则(1)f '= ； 8、曲线x y xe =的拐点是 ； 9、 2 1x dx -? = ； 10、设32,a i j k b i j k λ=+-=-+r r r r r r r r ，且a b ⊥r r ，则λ= ； 11、2lim()01x x ax b x →∞--=+，则a = ，b = ； 12、 3 11 lim x x x -→= ； 13、设()f x 可微，则 ()()f x d e = 。 二、 计算下列各题（每题5分，共20分） 1、 011 lim()ln(1)x x x →-+ 2 、y =y '； 3、设函数()y y x =由方程 xy e x y =+所确定，求0x dy =； 4、已知cos sin cos x t y t t t =?? =-?，求dy dx 。 三、 求解下列各题（每题5分，共20分） 1、4 21x dx x +? 2、2 sec x xdx ?
3 、 40? 4 、2201 dx a x + 四、 求解下列各题（共18分）： 1、求证：当0x >时， 2 ln(1)2x x x +>- （本题8分） 2、求由,,0x y e y e x ===所围成的图形的面积，并求该图形绕x 轴旋转一周所形成的旋转 体的体积。（本题10分） 高等数学（上）模拟试卷二 一、填空题（每空3分，共42分） 1 、函数 lg(1)y x =-的定义域是 ； 2、设函数 sin 0()20 x x f x x a x x ?
一、单选题（共15分，每小题3分） 1．设函数(,)f x y 在00(,)P x y 的两个偏导00(,)x f x y ，00(,)y f x y 都存在，则 ( ) A ．(,)f x y 在P 连续 B ．(,)f x y 在P 可微 C ． 0 0lim (,)x x f x y →及 0 0lim (,)y y f x y →都存在 D ． 00(,)(,) lim (,)x y x y f x y →存在 2．若x y z ln =，则dz 等于（ ）． ln ln ln ln .x x y y y y A x y + ln ln .x y y B x ln ln ln .ln x x y y C y ydx dy x + ln ln ln ln . x x y y y x D dx dy x y + 3．设Ω是圆柱面2 2 2x y x +=及平面01,z z ==所围成的区域，则 (),,(=???Ωdxdydz z y x f ） ． 21 2 0cos .(cos ,sin ,)A d dr f r r z dz π θ θθθ? ? ? 212 00 cos .(cos ,sin ,)B d rdr f r r z dz π θ θθθ? ? ? 212 2 cos .(cos ,sin ,)C d rdr f r r z dz π θ πθθθ-?? ? 21 0cos .(cos ,sin ,)x D d rdr f r r z dz π θθθ?? ? 4． 4．若 1 (1) n n n a x ∞ =-∑在1x =-处收敛，则此级数在2x =处（ ）． A ． 条件收敛 B ． 绝对收敛 C ． 发散 D ． 敛散性不能确定 5．曲线22 2 x y z z x y -+=?? =+?在点（1，1，2）处的一个切线方向向量为（ ）. A. （-1，3，4） B.（3，-1，4） C. （-1，0，3） D. （3，0，-1） 二、填空题（共15分，每小题3分） 1．设220x y xyz +-=，则' (1,1)x z = . 2．交 换ln 1 (,)e x I dx f x y dy = ? ? 的积分次序后，I =_____________________． 3．设2 2z xy u -=，则u 在点)1,1,2(-M 处的梯度为 . 4. 已知0!n x n x e n ∞ ==∑，则x xe -= . 5. 函数3322 33z x y x y =+--的极小值点是 . 三、解答题（共54分，每小题6--7分） 1.（本小题满分6分）设arctan y z y x =, 求z x ??,z y ??. 2.（本小题满分6分）求椭球面222 239x y z ++=的平行于平面23210x y z -++=的切平面方程，并求切点处的 法线方程. 3. （本小题满分7分）求函数2 2 z x y =+在点(1,2) 处沿向量122 l i j =+ r r r 方向的方向导数。 4. （本小题满分7分）将x x f 1 )(=展开成3-x 的幂级数，并求收敛域。 5．（本小题满分7分）求由方程088222 22=+-+++z yz z y x 所确定的隐函数),(y x z z =的极值。 6．（本小题满分7分）计算二重积分 1,1,1,)(222 =-=--=+??y y y x D d y x D 由曲线σ及2-=x 围成.
中国传媒大学 2009-2010学年第 一 学期期末考试试卷（B 卷） 及参考解答与评分标准 考试科目： 高等数学A （上） 考试班级： 2009级工科各班 考试方式： 闭卷 命题教师： 本大题共3小题，每小题3分，总计9分 ） 1、0)(0='x f 是可导函数)(x f 在0x 点处取得极值的 必要 条件。 2、设 )20() 1tan(cos ln π <
)(B 每个不定积分都可以表示为初等函数； )(C 初等函数的原函数必定是初等函数； )(D C B A ,,都不对。 答( D ) 3、若?-=x e x e dt t f dx d 0)(，则=)(x f x x e D e C x B x A 2222)( )()( )(----- 答( A ) 2小题，每小题5分，总计10分 ） 1、求极限0lim →x x x x 3sin arcsin -。 解：0lim →x =-x x x 3sin arcsin 0lim →x 3 arcsin x x x - （3分） lim →=x 31112 2=-- x x 0lim →x () ()x x x 621212 3 2---61-=。 （5分） 2、2tan ln x y =,求dx dy 。 解: 2sec 212 tan 12x x y ??= ' （3分） x x x x csc sin 1 2 cos 2sin 21==?=。 （5 分）
总复习(上) 一、求极限的方法： 1、利用运算法则与基本初等函数的极限； ①、定理 若lim (),lim ()f x A g x B ==, 则 (加减运算) lim[()()]f x g x A B +=+ (乘法运算) lim ()()f x g x AB = (除法运算) ()0,lim () f x A B g x B ≠=若 推论1: lim (),lim[()][lim ()]n n n f x A f x f x A === (n 为正整数) 推论2: lim ()[lim ()]cf x c f x = ②结论 结论2: ()f x 是基本初等函数，其定义区间为D ，若0x D ∈，则 0lim ()()x x f x f x →= 2、利用等价无穷小代换及无穷小的性质； ①定义1: 若0 lim ()0x x f x →=或（lim ()0x f x →∞=） 则称()f x 是当0x x → (或x →∞)时的无穷小. 定义2: ,αβ是自变量在同一变化过程中的无穷小: 若lim 1βα =, 则称α 与β是等价无穷小, 记为αβ . ②性质1：有限个无穷小的和也是无穷小. 性质2: 有界函数与无穷小的乘积是无穷小.
推论1: 常数与无穷小的乘积是无穷小. 推论2: 有限个无穷小的乘积也是无穷小. 定理2(等价无穷小替换定理) 设~,~ααββ'', 且lim βα'' 存在, 则 (因式替换原则) 常用等价无穷小: sin ~,tan ~,arcsin ~,arctan ~,x x x x x x x x ()()2 12 1cos ~ ,1~,11~,ln 1~,x x x e x x x x x μ μ--+-+ 1~ln ,x a x a -()0→x 3、利用夹逼准则和单调有界收敛准则； ①准则I(夹逼准则)若数列,,n n n x y z (n=1,2,…)满足下列条件: (1)(,,,)n n n y x z n ≤≤=123 ; (2)lim lim n n n n y z a →∞ →∞ ==, 则数列n x 的极限存在, 且lim n n x a →∞ =. ②准则II: 单调有界数列必有极限. 4、利用两个重要极限。 sin lim 1x x x →= 1 lim (1)x x x e →+= 1lim (1)x x e x →∞+ = 5、利用洛必达法则。 未定式为0,,,0,00∞ ∞∞-∞?∞∞ 类型.
2017-1-9 1. 设11ln(1)10()0x x x f x e x -+-<≤??=??>?，求()f x 的间断点，并指出间断点的类型。 0x =为跳跃间断点； 1x =为第二类间断点。 2.求,a b 的值，使点（1, 3）为曲线32y ax bx =+的拐点。 39,22 a b =-=。 3.已知两曲线()y f x =与2arctan 0 x t y e dt -=?在点(0, 0)处的切线相同，写出此切线方程，并求极限2lim ()n nf n →∞ 解 2()(0)2l i m ()l i m 2(0)2 1n n f f n nf f n n →∞→∞ -'=== 4. 求定积分 1 换元1x u = 5.求不定积分2x xe dx -? 221124 x x e x e C --=--+。 6.计算反常积分 21(1)dx x x +∞+? 11lim ln 2ln 222x →+∞== 7. 已知arctan x y t t ??=?=-??，求22d y dx 解 221111dy t t t dx t - +==+ 2222111d y t t dx t t +==+ 8. 判断级数1(1)(0)n n a n n a n ∞ +=+>∑的敛散性。
(1) 1l i m l i m (1)1n n a n n n a n n e n n +→∞→∞+=+= ， 所以，当1a >时，级数1 ()n n a n n a n ∞+=+ ∑收敛； 当01a <≤时，级数1 ()n n a n n a n ∞+=+∑发散。 9.设函数()y f x =在0x =的某邻域内具有一阶连续导数，且(0)0,(0)0f f '≠≠， 若()(2)(0)af h bf h f +-在0h →时是比h 高阶的无穷小，试确定,a b 的值 2,1a b ==-。 二、（9分）求极限111393lim 24(2)n n n →∞??? 11133934 lim 24(2)2n n n →∞???=。 三、（9分）设0A >，D 是由曲线段sin (0)2y A x x π=≤≤及直线0,2 y x π==所围成的平面区域，12,V V 分别表示D 绕x 轴与y 轴旋转所成旋转体的体积，若12V V =，求A 的值。 8A π= 四、（9分）设1 30()lim 1x x f x x e x →??++=???? ，其中f (x )在x = 0处二阶可导，求f (0)，f '(0)，f ''(0)。 (0)0f = 0()(0)lim (0)0x f x f f x →-'== f ''(0) = 4 五、(9分) 越野赛在湖滨举行，场地如图，出发点在陆地A 处，终点在湖心B 处，A ，B 南北相距5km ，东西相距7km ，湖岸位于A 点南侧2km, 是一条东西走向的笔直长堤。比赛中运动员可自行选择路线，但必须先从A 出发跑步到达长堤，再从长堤处下水游泳到达终点B 。已知某运动员跑步速度为118/v km h =，游泳速度为26/v km h =，问他应该在长堤的何处下水才能使比赛用时最少？ 6x = 此驻点是唯一的，则在点(6,0)R 下水，用时最少。
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 →-+-+-233 21216 29124 2、(本小题5分) .d )1(2 2x 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 21 6、(本小题5分) ??. d csc cot 46x x x 求 7、(本小题5分) ． 求? ππ 212 1cos 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 (),2 2 9、(本小题5分) ． 求dx x x ?+30 1 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 +20 2 sin 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
高等数学I 1. 当0x x →时，()(),x x αβ都是无穷小，则当0x x →时（ D ）不一定是 无穷小. (A) ()()x x βα+ (B) ()()x x 2 2βα+ (C) [])()(1ln x x βα?+ (D) )() (2x x βα 2. 极限a x a x a x -→??? ??1sin sin lim 的值是（ C ）. （A ） 1 （B ） e （C ） a e cot （D ） a e tan 3. ??? ??=≠-+=001 sin )(2x a x x e x x f ax 在0x =处连续，则a =（ D ）. （A ） 1 （B ） 0 （C ） e （D ） 1- 4. 设)(x f 在点x a =处可导，那么=--+→h h a f h a f h )2()(lim 0（ A ）. （A ） )(3a f ' （B ） )(2a f ' (C) )(a f ' （D ） ) (31 a f ' 二、填空题（本大题有4小题，每小题4分，共16分） 5. 极限) 0(ln )ln(lim 0>-+→a x a a x x 的值是 a 1. 6. 由x x y e y x 2cos ln =+确定函数y (x )，则导函数='y x xe ye x y x xy xy ln 2sin 2+++ - . 7. 直线l 过点M (,,)123且与两平面x y z x y z +-=-+=202356,都平行，则直 线l 的方程为 13 1211--=--=-z y x . 8. 求函数2 )4ln(2x x y -=的单调递增区间为 （－∞，0）和（1，+∞ ） . 三、解答题（本大题有4小题，每小题8分，共32分） 9. 计算极限10(1)lim x x x e x →+-.