Teacher: Liubiyu

Chapter 1-2

Contents §1.2 Elementary functions and graph §2.1 Limits of Sequence of number §2.2 Limits of functions §1.1 Sets and the real number §2.3 The operation of limits §2.4 The principle for existence of limits §2.5 Two important limits §2.6 Continuity of functions §2.7 Infinitesimal and infinity quantity, the order for infinitesimals

Purpose of teaching (1) To understand the concept of a function, to know the ways of representing a function and how to set the functional relationships based on the practical problem; (2) To know the bounded functions, monotone functions, odd function and even function, periodic functions (3) To understand the concept of composition of functions and piecewise functions, to know the concept of inverse functions and implicit functions

Purpose of teaching (4) To master properties of basic elementary functions and graph (5) To know how to construct function represent about simple application problems

New Words 递增 值域 递减 increasing 递增 range 值域 decreasing 递减 自变量 单调的 independent variable 自变量 monotonic 单调的 因变量 函数 dependent variable 因变量 functions 函数 定义域 奇函数 domain of definition 定义域 odd functions 奇函数 偶函数和 even functions 偶函数 sum 和 差 积分学 difference 差 integration 积分学 微积分 微分学 calculus 微积分 differentiation 微分学

复合函数积 composite functions 复合函数 product 积 分段函数 piecewise defined function 分段函数 反函数 商 inverse functions 反函数 quotient 商 初等函数 elementary functions 初等函数 隐函数 幂函数 implicit functions 隐函数 power functions 幂函数 指数函数 exponential functions 指数函数 对数函数 logarithm functions 对数函数 三角函数 trigonometric functions 三角函数 反三角函数 inverse trigonometric functions 反三角函数

§1.2 Elementary functions and graph This section develops the notion of a function, and shows how functions can be built up from simpler functions. 1 、 The concept of a function Definition 1 Definition 1

X function f x y Y Domain Range

Notations (1) The domain of definition and rule are the two important factors to determine the function. The former describes the region of existence of the function, and the latter gives the method for determining the corresponding elements of the set Y from the elements of the set X. A function is completely determined by these two factors and is independent of the forms of the expression and the kind of elements contained in the set.

(2) When the function is given by a formula, the domain is usually understood to consist of all the numbers for which the formula is defined. Example 1 Solution

Example 2 Solution

Example 3 Judge whether the following pair of functions are equal? Solution

2 、 Ways of representing a function To express a function is mainly to express its corresponding rule. There are many methods to express the corresponding rule, the following three are often used. (1) Analytic representation Many functions are given by an analytic representation. For example, the functions are given in example 1 and 2. (2) Method of tabulation Sometimes, a function is given by a table that lists the independent variable and its corresponding dependent variable. For example

(3) Method shown by graph The relation between y and x is shown by a graph. For example, the temperature curve recorded by some instruments expresses the relation between the temperature and time. t T

3 、 Properties of functions (1) Bounded Functions (2) Monotone Functions

(3) Odd Functions and Even Functions (3) Odd Functions and Even Functions

Notations (3) The graph of an odd function is symmetric with respect to the origin, and graph of an even function is symmetric with respect to the y-axis.

(3) Periodic Functions 2 T  2 3T  2 3T 2 T

Example 4 Solution

Example 5 Proof

4 、 Operation rule for functions (1) Rational operation rule for functions Definition 2 Definition 2

Definition 3 Definition 3 (composition of functions) (2) Composition of operations rule for functions

Notations (5) The above figure depicts the notion of a composite function.

Example 6 Solution Example 7 Example 7

Solution (3) inverse functions Definition 4 Definition 4 ( inverse functions)

Notations

)(xfy  x y o ),(abQ ),(baP

Example 8 Example 8 Find the inverse function of the following functions Solution

5 、 Piecewise defined function It should be noted that the analytic representation of a function sometimes consists of several components on different subsets of the domain of definition of the function. A function expressed by this kind of representations is called a piecewise defined function.

Example 9 Example 9 ( Sign function ) Example 10 ( The greatest integer function )

( The Dirichlet’s function ) Example 11 Example 11 Example 12 Example 12 ( Integer variable function )

Solution Example 13 Example 13

6 、 Elementary functions

Definition 5 Definition 5 ( elementary function ) A function formed from the six kinds of basic elementary functions by a finite number of rational operations and compositions of functions which can be expressed by a single analytic expression is called an elementary functions. Notations

Definition 6 Definition 6 ( Hyperbolic function )

There are some identities for hyperbolic functions which are similar to those for trigonometric functions.

Notations

7 、 Implicit functions

Definition 7