微积分大一基础知识经典讲解.pdf
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 domain. )()(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 functions 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 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 : 022 ,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 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 coefficient . 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 xfExample 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 )()( It 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) 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 to a(on either side of a) but not equal to a. Note axmeans 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 fis defined near a. 2.Limit Laws Suppose that c is a constant and the limits)(limand)(limxgxf axax exist. Then )(lim)(lim)()(lim)1xgxfxgxf axaxax )(lim)(lim)()(lim)2xgxfxgxf axaxax 0)(lim )(lim )(lim )( )( lim)3xgif xg xf xg xf ax ax ax ax Note From 2), we have )(l i m)(l i mxfcxcf axax integer.positiveais,)(lim)(limnxfxf n ax n ax 3. 1) 2) Note 4.One-Sided Limits 1)left-hand limit Definition We write Lxf ax )(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 Lxf ax )(lim and say “f(x) tends to L as x tends to a from right ” if we can make the values 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)(limxfLxfLxf axaxax |limFind 0 x x Example1 Solution x x x | limFind 0 Example2 Solution 6.Infinitesimals(无穷小量 ) and infinities(无穷大量 ) 1)Definition 0)(limxf x We say f(x) is an infinitesimal as where,xis some number or . Example1 22 0 0limxx x is an infinitesimal as .0x Example2 xx x 1 0 1 limis an infinitesimal as.x 2)Theorem 0)(limxf x and g(x) is bounded.0)()(limxgxf x Note Example 0 1 sinlim 0 x x x 3)Definition )(limxf x We say f(x) is an infinity as where,xis some number or . Example1 1 1 1 1 lim 1 xx x is an infinity as.1x Example2 22 limxx x is an infinity as.x 4)Theorem 0 )( 1 lim)(lim) xf xfa xx )( 1 limat possiblyexcept near0)(, 0)(lim) xf xfxfb xx 13 124 lim 4 23 x xx x Example1 4 42 1 3 124 lim x xxx x 13 322 lim 2 2 n nn n Example2 2 2 1 3 32 2 lim n nn n 3 2 xx x x 78 12 lim 2 3 Example3 2 3 78 1 2 lim xx x x Note mnif mnif mnif b a bxbxb axaxa n n m m m m n n n n x 0lim 0 1 1 0 1 1 , 0, 0andconstantsare), 0(), 0(where 00 bamjbnia ji m, n are nonnegative integer. Exercises )6(),0(3 12 2 lim)1.1 2 ba n bnan n ) 1(),1(1) 1 (lim)2 2 babax x x x )2(),2(2 1 lim)3 1 ba x bax x 4 3 14 3 lim)1.2 2 2 n nn n 5 1 )2(5 )2(5 lim)2 11nn nn n 3 4 3 1 3 1 1 2 1 2 1 1 lim)3 n n n 1) 1231 (lim)4 222 n n nn n 1) )1( 1 32 1 21 1 (lim)5 nn n 2 1 )1(lim)6nnn n 4 43 lim)1 .3 2 2 2 x xx x 2 33 0 3 )( lim)2x h xhx h 3 43 153 l i m)3 2 2 xx xx x 50 3020 50 3020 5 32 ) 15( )23()32( lim)4 x xx x 2) 1 2)( 1 1(lim)5 2 xx x 0 724 132 lim)6 5 3 xx xx x 4 2 1 13 lim)7 2 1 x xx x 1) 1 3 1 1 (lim)8 3 1 xx x 3 2 1 1 lim)9 3 1 x x x 6 1)31)(21)(1( lim)10 0 x xxx x 2 1 )1)(2(lim)11xxx x 2 2 3 )3( 3 lim) 1.4 x xx x 43 2 lim)2 3 x x x )325(lim)3 2 xx x 1)2544(lim.5 2 xxx x
