Chapter1 Functions(函数)1.Definition 1)A function 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 domain.⇔=)()(x g x f :N ote1)(,11)(2+=--=x x g x x x f Example )()(x g x f ≠⇒2.Basic Elementary Functions(基本初等函数) 1) constant functions f (x )=c2) power functions0,)(≠=a x x f a3) exponential functions1,0,)(≠>=a a a x f xdomain: R range: ),0(∞4) logarithmic functions1,0,log)(≠>=a a x x f adomain: ),0(∞ range: R5) trigonometric functionsf (x )=sin x f (x )=cos x f (x )=tan x f (x )=cot x f (x )=sec x f (x )=csc xGiven two functions f and g , the composite function(复合函数) g f is defined by))(())((x g f x g f =Note )))((())((x h g f x h g f =Example If ,2)()(x x g and x x f -==find each function and its domain.gg d ff c fg b gf a ))))))(())(()x g f x g f a = Solution )2(x f -=422xx -=-=]2,(}2{:domain -∞≤or x xxx g x f g x f g b -===2)())(())(()]4,0[:02,0domain x x ⇒⎩⎨⎧≥-≥ 4)())(())(()xx x f x f f x f f c ==== )[0, :domain ∞xx g x g g x g g d --=-==22)2())(())(()]2,2[:022,02-⇒⎩⎨⎧≥--≥-domain 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+=x x F is an elementary function.)))((()()(cos )(9)(2x h g f x F xx f xx g x x h ===+=2sin 1log)(xex x f xa-+=Example is an elementary function.1)Polynomial(多项式) FunctionsRx a x a xa x a x P n n nn ∈++++=--0111)( where n is a nonnegative integer.The leading coefficient(系数) ⇒≠.0n a The degree of the polynomial is n . In particular(特别地),The leading coefficient ⇒≠.00a constant function The leading coefficient ⇒≠.01a linear functionThe leading coefficient ⇒≠.02a quadratic(二次) function The leading coefficient ⇒≠.03a cubic(三次) function2)Rational(有理) Functions}.0)(such that is {,)()()(≠=x Q x x x Q x P x f where P and Q are polynomials.3) Root Functions4.Piecewise Defined Functions(分段函数)⎩⎨⎧>≤-=111)(x if x x if x x f Example5.6.Properties(性质) 1)Symmetry(对称性)even function : x x f x f ∀=-),()( in its domain.symmetric w.r.t.(with respect to 关于) the y -axis.odd function : x x f x f ∀-=-),()( in its domain. symmetric about the origin.2) monotonicity(单调性)A function f is called increasing on interval(区间) I if I in x x x f x f 2121)()(<∀< It is called decreasing on I if I in x x x f x f 2121)()(<∀> 3) boundedness(有界性)belowbounded )(xex f =Example1abovebounded )(xex f -=Example2belowand above from bounded sin )(x x f =E xample34) periodicity (周期性) Example f (x )=sin xChapter 2 Limits and Continuity1.Definition We write L x f ax =→)(limand 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 a x ≠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 LawsSuppose that c is a constant and the limits )(lim and )(lim x g x f ax ax →→exist. Then)(lim )(lim )]()([lim )1x g x f x g x f ax ax ax →→→±=±)(lim )(lim )]()([lim )2x g x f x g x f ax ax ax →→→⋅=0)(lim )(lim )(lim )()(lim)3≠=→→→→x g if x g x f x g x f ax ax ax axNote From 2), we have )(lim )(lim x f c x cf ax ax →→=integer. positive a is ,)](lim [)]([lim n x f x f nax n ax →→=3. 1) 2)Note4.One-Sided Limits 1)left-hand limitDefinition We write L x f ax =-→)(limand 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 limitDefinition We write L x f ax =+→)(limand 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 )(lim x f L x f L x f ax ax ax +-→→→==⇔=||lim Find 0x x → E xample1Solutionxx x ||limFind 0→ Example2Solution6.Infinitesimals(无穷小量) and infinities(无穷大量)1)Definition ⇒=∆→0)(lim x f x We say f (x ) is an infinitesimal as ∆∆→ where ,x issome number or .∞±Example1 2200lim x x x ⇒=→ is an infinitesimal as .0→xExample2 xxx 101lim⇒=±∞→ is an infinitesimal as .±∞→x2)Theorem 0)(lim =∆→x f x and g(x) is bounded.0)()(lim =⇒∆→x g x f xNoteExample 01sinlim 0=→xx x3)Definition ⇒±∞=∆→)(lim x f x We say f (x ) is an infinity as ∆∆→ where ,x is somenumber or .∞± Example1 1111lim1-⇒∞=-+→x x x is an infinity as .1+→xExample2 22lim x x x ⇒∞=∞→ is an infinity as .∞→x4)Theorem0)(1lim)(lim )=⇒±∞=∆→∆→x f x f a x x±∞=⇒∆∆≠=∆→∆→)(1limat possiblyexcept near 0)(,0)(lim )x f x f x f b x x13124lim423+-+∞→x x x x Example144213124limx xxxx +-+=∞→ 0=13322lim22++-∞→n n n n Example2 2213322limnnn n ++-=∞→ 32=xx x x 7812lim23++∞→E xample3 237812limxxxx ++=∞→ ∞= Note ⎪⎪⎪⎩⎪⎪⎪⎨⎧>∞<==++++++-----∞→m n if m n if m n if b a b xb xb a x a x a n nm m mm n n n n x 0lim11011 ,0,0and constants are ),,0(),,,0(where 00≠≠==b a m j b n i a j i m , n arenonnegative integer.Exercises)6(),0(3122lim)1.12==⇒=-++∞→b a n bn ann)1(),1(1)1(lim )22-==⇒=--+∞→b a b ax xx x)2(),2(21lim)31-==⇒=-+→b a x b ax x43143lim)1.222=++∞→n n n n 51)2(5)2(5lim)211=-+-+++∞→n n n n n343131121211lim)3=++++++∞→nnn 1)1231(lim )4222=-+++∞→nn n n n 1))1(1321211(lim )5=+++∙+∙∞→n n n 21)1(lim )6=-+∞→n n n n∞=---→443lim)1.3222x x x x 23303)(lim)2x hxh x h =-+→343153lim)322=++++∞→x x x x x 503020503020532)15()23()32(lim)4∙=+++-∞→x x x x2)12)(11(lim )52=-+∞→xxx 0724132lim)653=++++∞→x x x x x42113lim)721-=-+--→x xx x 1)1311(lim )831-=---→xxx3211lim)931=--→x x x 61)31)(21)(1(lim)100=-+++→xx x x x21))1)(2((lim )11=--++∞→x x x x∞=-+→223)3(3lim)1.4x x x x ∞=++∞→432lim)23x x x∞=+-∞→)325(lim )32x x x1)2544(lim .52-=+++-∞→x x x x。