2023年微积分大一基础知识经典讲解.pdf

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1、微积分大一基础知识经典讲解 Chapter1 Functions(函数)1、Definition 1)Afunction f is a rule that assigns to each element x in a set A exactly one element,called f(x),in a set B、2)The set A is called the domain(定义域)of the function、3)The range(值域)of f is the set of all possible values of f(x)as x varies through out the

2、domain、)()(xgxf:Note 1)(,11)(2xxgxxxfExample)()(xgxf 2、Basic Elementary Functions(基本初等函数)1)constant functions f(x)=c 2)power functions 0,)(axxfa 3)exponential functions 1,0,)(aaaxfx domain:R range:),0(4)logarithmic functions 1,0,log)(aaxxfa domain:),0(range:R 5)trigonometric functions f(x)=sinx f(x)

3、=cosx f(x)=tanx f(x)=cotx f(x)=secx f(x)=cscx 6)inverse trigonometric functions domain range graph f(x)=arcsinx or x1sin 1,1 2,2 f(x)=arccos x or x1cos 1,1,0 f(x)=arctanx or x1tan R)2,2(f(x)=arccotx or x1cot R),0(3、Definition Given two functions f and g,the composite function(复合函数)gf is defined by)(

4、)(xgfxgf Note)()(xhgfxhgf 微积分大一基础知识经典讲解 Example If,2)()(xxgandxxf find each function and its domain、ggdffcfgbgfa)()()xgfxgfaSolution)2(xf422xx 2,(2:domainorxx xxgxfgxfgb2)()()()4,0:02,0domainxx 4)()()()xxxfxffxffc)0,:domain xxgxggxggd22)2()()()2,2:022,02domainxx 4、Definition An elementary function(初

5、等函数)is constructed using combinations(addition 加,subtraction 减,multiplication 乘,division 除)and composition starting with basic elementary functions、Example)9(cos)(2xxF is an elementary function、)()()(cos)(9)(2xhgfxFxxfxxgxxh 2sin1log)(xexxfxaExample is an elementary function、1)Polynomial(多项式)Functio

6、ns RxaxaxaxaxPnnnn0111)(where n is a nonnegative integer、The leading coefficient(系数).0naThe degree of the polynomial is n、In particular(特别地),The leading coefficient .00aconstant function The leading coefficient .01alinear function The leading coefficient .02aquadratic(二次)function The leading coeffic

7、ient .03acubic(三次)function 微积分大一基础知识经典讲解 2)Rational(有理)Functions .0)(such that is,)()()(xQxxxQxPxf where P and Q are polynomials、3)Root Functions 4、Piecewise Defined Functions(分段函数)111)(xifxxifxxfExample 5、6、Properties(性质)1)Symmetry(对称性)even function:xxfxf),()(in its domain、symmetric w、r、t、(with res

8、pect to 关于)the y-axis、odd function:xxfxf),()(in its domain、symmetric about the origin、2)monotonicity(单调性)A function f is called increasing on interval(区间)I if Iinxxxfxf2121)()(It is called decreasing on I if Iinxxxfxf2121)()(3)boundedness(有界性)below bounded)(xexfExample1 above bounded)(xexfExamp le2

9、below and above from boundedsin)(xxfExample3 4)periodicity(周期性)Example f(x)=sinx Chapter 2 Limits and Continuity 1、Definition We write Lxfax)(lim and say“f(x)approaches(tends to 趋向于)L as x tends to a”if we can make the values of f(x)arbitrarily(任意地)close to L by taking x to be sufficiently(足够地)close

10、 to a(on either side of a)but not equal to a、Note ax means that in finding the limit of f(x)as x tends to a,we never consider x=a、In fact,f(x)need not even be defined when x=a、The only thing that matters is how f is defined near a、2、Limit Laws Suppose that c is a constant and the limits)(limand)(lim

11、xgxfaxaxexist、Then 微积分大一基础知识经典讲解)(lim)(lim)()(lim)1xgxfxgxfaxaxax)(lim)(lim)()(lim)2xgxfxgxfaxaxax 0)(lim)(lim)(lim)()(lim)3xgifxgxfxgxfaxaxaxax Note From 2),we have )(lim)(limxfcxcfaxax integer.positive a is,)(lim)(limnxfxfnaxnax 3、1)2)Note 4、One-Sided Limits 1)left-hand limit Definition We write L

12、xfax)(lim and say“f(x)tends to L as x tends to a from left”if we can make the values of f(x)arbitrarily close to L by taking x to be sufficiently close to a and x less than a、2)right-hand limit Definition We write Lxfax)(lim and say“f(x)tends to L as x tends to a from right”if we can make the values

13、 of f(x)arbitrarily close to L by taking x to be sufficiently close to a and x greater than a、5、Theorem)(lim)(lim)(limxfLxfLxfaxaxax|limFind0 xx Example1 Solution xxx|limFind0 Example2 Solution 6、Infinitesimals(无穷小量)and infinities(无穷大量)1)Definition 0)(limxfxWe say f(x)is an infinitesimal as where,x

14、is some number or.Example1 2200limxxx is an infinitesimal as.0 x 微积分大一基础知识经典讲解 Example2 xxx101lim is an infinitesimal as.x 2)Theorem 0)(limxfx and g(x)is bounded、0)()(limxgxfx Note Example 01sinlim0 xxx 3)Definition)(limxfxWe say f(x)is an infinity as where,x is some number or.Example1 1111lim1xxx i

15、s an infinity as.1x Example2 22limxxx is an infinity as.x 4)Theorem 0)(1lim)(lim)xfxfaxx)(1limat possiblyexcept near0)(,0)(lim)xfxfxfbxx 13124lim423xxxxExample1 44213124limxxxxx 0 13322lim22nnnnExample2 2213322limnnnn 32 xxxx7812lim23Example3 237812limxxxx Note mnifmnifmnifbabxbxbaxaxannmmmmnnnnx0li

16、m011011,0,0and constants are),0(),0(where00bamjbniajim,n are nonnegative integer、Exercises 微积分大一基础知识经典讲解)6(),0(3122lim)1.12banbnann)1(),1(1)1(lim)22babaxxxx)2(),2(21lim)31baxbaxx 43143lim)1.222nnnn 51)2(5)2(5lim)211nnnnn 343131121211lim)3nnn 1)1231(lim)4222nnnnn 1)1(1321211(lim)5nnn 21)1(lim)6nnnn 443lim)1.3222xxxx 23303)(lim)2xhxhxh 343153lim)322xxxxx 503020503020532)15()23()32(lim)4xxxx 2)12)(11(lim)52xxx 0724132lim)653xxxxx 42113lim)721xxxx 1)1311(lim)831xxx 3211lim)931xxx 61)31)(21)(1(lim)100 xxxxx 21)1)(2(lim)11xxxx 223)3(3lim)1.4xxxx 432lim)23xxx)325(lim)32xxx 1)2544(lim.52xxxx