参考分析成果chapter

1、1Happy study! 王平平 教授/博士 江西财经大学统计学院 Tel: Email: 21 Functions2 Limits and Derivatives3 The Differentiation Rules4 Applications of Differentiation5 integrals*The best way to learn mathematics is to do mathematics.*Calculus (I)3Chapter 1 Functions1.Functions Definition 1 A function f is a rule that assi

2、gns to each element x in a set D exactly one element , called f(x), in a set R. x is called an independent variable and y (= f(x) is called a dependent variable. The set D is called the domain of the function f. The number f(x) is the value of f at x. The range (R) of f is the set of all possible va

3、lues of f(x) as x varies throughout the domain. The graph of the function f is the set of all point (x, f(x), where x is in D;i.e., the set (x, f(x)xD.4Example 1 Given a function f(x)=1-x2, find the domain and the range of f, and sketch its graph.Solution The domain of f is the set of all real numbe

4、rs such that f(x) is real. As f(x) exists if and only if the radicand 1-x2 is nonnegative; i.e., 1-x20, or equivalently, -1x1, the domain is -1, 1. As 01-x21, whenever x is in -1, 1,it follows that 01-x21. Hence , the range is 0, 1.The graph of f(x) is the upper portion of the unite circle as illust

5、rated in Figure 1(a). (a) Figure 1 (b)5678 Functions that are described by more than one expression, as in the next example, are called piecewise functions.Example 2 Sketch the graph of the function defined as follows: x+1, if x2.Solution If x2, the function values are a constant number 1, and the g

6、raph of f is a horizontal half-line without the endpoint (2, 1). 92. Some Properties of Functions2.1 Symmetry Let f be a function with domain D which is symmetric with respect to the origin (i.e.,xD implies that xD). If f(x)=f(-x) for all x in D, then f is called an even function. If f(-x)=-f(x) for

7、 all x in D, f is called an odd function.If f is an even function, the graph of f is symmetric with respect to the y-axis, since if a point(x, f(x) is on the graph, the point (-x, f(x) is also on the graph. If f is an odd function, the graph of f is symmetric with respect to the origin as for any x

8、in the domain of f, points (x, f(x) and (-x, -f(x) are both on the graph. 10 A typical example of even functions is the function f(x)=x2, since f(-x)=(-x)2=x2=f(x) for every xR . The function f(x)=x3+x is an example of odd functions according to the definition of odd function. The graph of these two

9、 functions are sketched in Figure 2. Figure 2112.2 Boundedness For a function f , if there exists a number M such that f(x)M for every x in S, where S is a subset of the domain of f , then f is said to be bounded above in S. The number M is called an upper boundary of f. If there is a number m such

10、that f(x)m for every x in S, f is said to be bounded below in S, and the number m is a lower boundary of f. A function f is bounded if it is bounded above and below in its domain. For example, the function y=sin x is a bounded function, since we can find a number 1 such that sinx1; i.e., -1sinx1, fo

11、r every real number x. Hence y=sin x has an upper boundary 1 and a lower boundary 1. Actually, any number greater than 1 could be an upper boundary of y=sinx and any number less than 1 could be a lower boundary. 122.3 Monotonicity A function f is called increasing on an interval I if f(x1)f(x2)whene

12、ver x1f(x2) whenever x1x2 in I. For instance, the function y=x2 is increasing in the interval 0, +) and decreasing in (-, 0. But it is neither increasing nor decreasing in its domain. The function y=x3+x is increasing in its domain. (see Figure 2) In general, if a function is increasing in its domai

13、n, it is called an increasing function ; if a function is decreasing in its domain, it is called a decreasing function. 13 A function f is said to be periodic if there exists a real number T such that whenever x is in the domain of f, then x+T is also in the domain of f and f(x+T)=f(x). The least po

14、sitive number T is called the period of f. For example, the function y=cosx is a period function and 2 is its period.3. Composite Functions and Inverse Functions3.1 Composite FunctionsDefinition 2 The composite function fg (also called the composition of f and g) is defined by (fg)(x)=f(g(x). The do

15、main of fg is the set of all x in the domain of g such that g(x) is in the domain of f. 14 A sufficient and necessary condition of forming fg is that the intersection of the range of g and the domain of f is not empty.Example 1 If f(x)=1/(1-x2) and g(x)=x, find1 (fg)(x) and the domain of fg,2 (gf)(x

16、) and the domain of gf.Solution We first notice that the domain of f is the set (-, -1)(-1, 1)(1, +) and the domain of g is the interval 0, +).1 (fg)(x)=f(g(x)=f(x)=1/(1-x). The domain of fg is the set 0, 1)(1, +).2 (gf)(x) =g(f(x)=g(1/(1-x2)=1/(1-x2). The domain of gf is the open interval (-1, 1).

17、15Example 2 Find fgh if h(x)=lnx, g(x)=cotx and f(x)=1+x2.Solution By definition, (fgh)(x)=f(g(h(x)=f(g(lnx)=f(cot(lnx)=1+(cot(lnx)2. In calculus it is sometimes useful to be able to pose a complicated function into simpler ones, see the following example.Example 3 Given h(x)=ex+sinx, find function

18、f and g such that h=fg .Solution We let g(x)=x+sinx, and f(x)=ex, then (fg)(x)=f(g(x)=f(x+sinx)=ex+sinx=h(x).163.2 Inverse Functions Definition 3 Let f be a one-to-one function with domain X and range Y. A function g with domain Y and range X is called inverse function of f if g satisfies the condit

19、ion: g(y)=x f(x)=y , for any y in Y. The symbol f-1 is often used to denote the inverse function g of f . According to Definition 3, we must have x=f-1(y). If we interchange x and y, we get y=f-1(x), which is the desired function.17The graph of y = f -1(x).18The graph of y = f -1(x).19Example 4 Find

20、 the inverse function of f(x)=2x-1.Solution For the function f(x)=2x-1, we first write y=2x-1.Then we solve this equation for x in terms of y and obtain x=(y+1)/2. Finally interchange x and y yields that y=(x+1)/2. Thus the inverse function is f-1(x)=(x+1)/2. For instance, the function f(x)=x2 is no

21、t a one-to-one function on its domain R. Hence it has no any inverse function on R. But if f is restricted on the set of nonnegative real numbers, f is one-to-one and has the inverse function f-1(x)=x .20 y=sinx, y=arcsinx y=cosx, y=arccosx y=x2, y=x y=tanx, y=arctanx21 4. Elementary Functions Eleme

22、ntary Functions are commonly occurring functions in calculus.4.1 Basic elementary function (1) Constant function f(x)=c (c=constant) , where c is a real number. y c o x Figure 1*The best way to learn mathematics is to do mathematics.* 22 (2) Power functions A function of the form , where a is a cons

23、tant,is called a power function. (3) Exponential functions An exponential function is a function of the form ,where a is a positive constant, and a1. The domain of f(x) is R and the range is (0, + ).If 0a1, f(x) is an increasing function. y a1 1 o x Figure 2 23 (4) Logarithmic functionsA logarithmic

24、 function to the base a is a function of the form ， where a0 and a1. (5) Trigonometric functions There are six basic types of trigonometric functions: y=sinx, y=cosx, y=tanx, y=cotx,y=secx, and y=cscx. (6) Anti-trigonometric functions There are six basic types of anti-trigonometric functions:y=arcsinx, y=arccosx, y=arctanx, y=arccotx, y=arcsecx, and y=arccscx. They are inverse functions of trigonometric functions respectively.24The graphs of some functions are shown in Figure 3. y=sinx, y=cosx y=tanx, y=cotx y=arcsinx, y=arccosx y=ex, y=lnx Figure 3 Figure 3254.2