宁波大学《高等数学C1》教学-Chapter 00 Preliminaries.pptVIP

Chapter 1 Functions 1.3 Basic Elementary Functions 1.1 The Conception of Functions 1.2 Some Properties of Functions 3. Pairity(奇偶性) 4. Periodicity（周期性）[?piri:??disiti:] 5. Continuity（连续性） 1.3 Basic Elementary Functions 1.4 Composite Functions 1.5 Elementary Functions 非初等函数举例: Exercises P22 41 P42 24,28 P81 10, 11, 19, * 目录 上页 下页 返回 结束 1.1 The Conception of Functions 1.2 Some Properties of Functions 1.4 Composite Functions 1.5 Elementary Functions 则 f , 使得 有唯一确定的 与之对应 , 则 称 f 为从 X 到 Y 的函数, 记作 Independent Variable Domain Dependent Variable 设 X , Y 是两个非空实数集合, 若存在一个对应规 Domian 使表达式或实际问题有意义的自变量集合. 对无实际背景的函数, 书写时可以省略定义域. 对实际问题, 书写函数时必须写出定义域； Function Representation: 解析法 、图像法 、列表法 EXAMPLE 1. Inverse Sine Function Domian Range Absolute Value Function Domian Range EXAMPLE 2. Say whether the following representations  is a function ？ Function Element Domian and f EXAMPLE 3. Say whether two functions are equivalent? Neighborhood (邻域) 其中, a 称为邻域中心 , ? 称为邻域半径 . 左邻域 : 右邻域 : Suppose 1. Boundedness (有界性) 使 则: 为D上的有界函数. 在D上有界； 几何解释（Geometry） 有上(下)界；充要条件（Necessary and Sufficient Condition） is bounded. EXAMPLE 1. Show that the function Solution. For any we have Is M single？ Whether it is easy to seek M？Other methods to the same purpose? Unbounded whereas 2. Monotonicity (单调性) then increases on D ; Further discussion… If If then decreases on D . then is non-decreasing on D ; If If then is non-increasing on D . EXAMPLE 1. Even Hyperbolic Cosine Hyperbolic Sine Odd Hyperbolic Tangent Odd Solution: , we have Even function Odd function EXAMPLE 2. Given an arbitrary function f(x), try to express it by the addition of an even function and an odd function. For any and ,then we call periodic function , if and l period. ( Usually refers to Minimum Positive Period ). Period Period Note: 周期函数不一定存在最小正周期 . EXAMPLE. 2. Dirichlet（狄利克雷函数） x 为有理数 x 为无理数 1. Constant Function 6. Derivability（可导性） 7. Different