微积分大一基础知识经典讲解.pdf
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1、Chapter1 Functions( 函数) 1.Definition 1)Afunctionf 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 domai
2、n. )()(xgxf:Note 1)(, 1 1 )( 2 xxg x x xfExample)()(xgxf 2.Basic Elementary Functions( 基本初等函数 ) 1) constant functions f(x)=c 2) power functions 0,)(axxf a 3) exponential functions 1,0,)(aaaxf x domain: R range: ), 0( 4) logarithmic functions 1,0,log)(aaxxf a domain: ),0(range: R 5) trigonometric fun
3、ctions f(x)=sinx f(x)=cosx f(x)=tanx f(x)=cotx f(x)=secx f(x)=cscx 6) inverse trigonometric functions domain range graph f(x)=arcsinx or x 1 sin 1, 1 2 , 2 f(x)=arccosx or x 1 cos 1, 1,0 f(x)=arctanx or x 1 tan R ) 2 , 2 ( f(x)=arccotx or x 1 cot R ),0( 3. Definition Given two functions f and g, the
4、 composite function( 复合函数 ) gfis defined by )()(xgfxgf Note )()(xhgfxhgf Example If ,2)()(xxgandxxffind each function and its domain. ggdffcfgbgfa) )()()xgfxgfaSolution)2(xf 4 22xx 2,(2:domainorxx xxgxfgxfgb2)()()() 4,0 : 02 ,0 dom ain x x 4 )()()()xxxfxffxffc)0,:domain xxgxggxggd22)2()()() 2,2 : 02
5、2 ,02 dom ain x x 4.Definition An elementary function( 初等函数 ) is constructed using combinations (addition 加, subtraction减, multiplication 乘, division 除) and composition starting with basic elementary functions. Example )9(cos)( 2 xxFis an elementary function. )()()(cos)(9)( 2 xhgfxFxxfxxgxxh 2 sin 1
6、 log)( x e xxf x a Exampleis anelementary function. 1)Polynomial(多项式 ) Functions RxaxaxaxaxP n n n n01 1 1 )(where n is a nonnegative integer. The leading coefficient(系数) .0 n aThe degree of the polynomial is n. In particular(特别地 ), The leading coefficient .0 0 aconstant function The leading coeffic
7、ient . 0 1 alinear function The leading coefficient .0 2 aquadratic(二次) function The leading coefficient .0 3 acubic(三次) function 2)Rational(有理) Functions .0)(such thatis, )( )( )(xQxx xQ xP xfwhere P and Q are polynomials. 3) Root Functions 4.Piecewise Defined Functions( 分段函数 ) 1 11 )( xifx xifx xf
8、Example 5. 6.Properties(性质) 1)Symmetry(对称性 ) even function : xxfxf),()(in its domain. symmetric w.r.t.(with respect 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 Iinxxxfxf 2121 )()( I
9、t is called decreasing on I if Iinxxxfxf 2121 )()( 3) boundedness( 有界性 ) belowbounded)( x exfExample1 abovebounded)( x exfExample2 belowandabovefromboundedsin)(xxfExample3 4) periodicity (周期性 ) Example f(x)=sinx Chapter 2 Limits and Continuity 1.Definition We write Lxf ax )(lim and say “f(x) approac
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