微积分大一基础知识经典讲解讲课稿

微积分大一基础知识经典讲解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 :Note1)(,11)(2+=--=x x g x x x f E xample )()(x g x f ≠⇒ 2.Basic Elementary Functions(基本初等函数)1) constant functionsf (x )=c2) power functions0,)(≠=a x x f a3) exponential functions1,0,)(≠>=a a a x f x domain: R range: ),0(∞4) logarithmic functions1,0,log )(≠>=a a x x f a domain: ),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.g g d f f c f g b g f a ))))))(())(()x g f x g f a = Solution )2(x f -=422x x -=-=]2,(}2{:domain -∞≤or x xx x g x f g x f g b -===2)())(())(()]4,0[:02,0domain x x ⇒⎩⎨⎧≥-≥ 4)())(())(()x x x f x f f x f f c ==== )[0, :domain ∞x x 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 compositionstarting with basic elementary functions.Example )9(cos )(2+=x x F is an elementary function.)))((()()(cos )(9)(2x h g f x F x x f x x g x x h ===+=2sin 1log )(x e x x f x a -+=E xample is an elementary function.1)Polynomial(多项式) FunctionsR x a x a x a x a x P n n n n ∈++++=--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 functionThe leading coefficient ⇒≠.01a linear functionThe leading coefficient ⇒≠.02a quadratic(二次) functionThe 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 xx if x x f Example 5.6.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)below bounded )(x e x f =E xample1above bounded )(x e x f -=E xamp le2below and above from bounded sin )(x x f =Example34) periodicity (周期性)Example f (x )=sin xChapter 2 Limits and Continuity1.Definition We write L x f 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 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 a x →→exist. Then )(lim )(lim )]()([lim )1x g x f x g x f ax a x a x →→→±=± )(lim )(lim )]()([lim )2x g x f x g x f ax a x a x →→→⋅= 0)(lim )(lim )(lim )()(lim )3≠=→→→→x g if x g x f x g x f a x ax a x a x Note From 2), we have)(lim )(lim x f c x cf ax a x →→= integer. positive a is ,)](lim [)]([lim n x f x f n ax n a x →→= 3.1)2)Note4.One-Sided Limits1)left-hand limitDefinition We write L x f 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 limitDefinition We write L x f 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 )(lim x f L x f L x f ax a x a x +-→→→==⇔= ||lim Find 0x x → Example1 Solutionxx x ||limFind 0→ Example2 Solution6.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 is some number or .∞±Example1 2200lim x x x ⇒=→ is an infinitesimal as .0→x Example2 xx x 101lim ⇒=±∞→ is an infinitesimal as .±∞→x 2)Theorem 0)(lim =∆→x f x and g(x) is bounded.0)()(lim =⇒∆→x g x f xNote Example 01sin lim 0=→xx x 3)Definition ⇒±∞=∆→)(lim x f x We say f (x ) is an infinity as ∆∆→ where ,x is some number or .∞± Example1 1111lim 1-⇒∞=-+→x x x is an infinity as .1+→x Example2 22lim x x x ⇒∞=∞→ is an infinity as .∞→x4)Theorem0)(1lim )(lim )=⇒±∞=∆→∆→x f x f a x x ±∞=⇒∆∆≠=∆→∆→)(1lim at possibly ex cept near 0)(,0)(lim )x f x f x f b x x 13124lim 423+-+∞→x x x x E xample1 44213124lim xx x x x +-+=∞→ 0= 13322lim 22++-∞→n n n n E xample2 2213322lim nn n n ++-=∞→ 32= x x x x 7812lim 23++∞→E xample3 237812lim xx x x ++=∞→ ∞=Note ⎪⎪⎪⎩⎪⎪⎪⎨⎧>∞<==++++++-----∞→m n if m n if mn if b a b x b x b a x a x a n n m m m m n n n n x 0lim 011011 ,0,0and constants are ),,0(),,,0(where 00≠≠==b a m j b n i a j i m , n are nonnegative integer. Exercises)6(),0(3122lim )1.12==⇒=-++∞→b a n bn an n )1(),1(1)1(lim )22-==⇒=--+∞→b a b ax xx x )2(),2(21lim )31-==⇒=-+→b a x b ax x 43143lim )1.222=++∞→n n n n 51)2(5)2(5lim )211=-+-+++∞→n n n n n 343131121211lim )3=++++++∞→n n n 1)1231(lim )4222=-+++∞→n n 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 h x h x h =-+→ 343153lim )322=++++∞→x x x x x 503020503020532)15()23()32(lim )4•=+++-∞→x x x x 2)12)(11(lim )52=-+∞→xx x 0724132lim )653=++++∞→x x x x x 42113lim )721-=-+--→x x x x 1)1311(lim )831-=---→x x x 3211lim )931=--→x x x 61)31)(21)(1(lim )100=-+++→x x 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 x 1)2544(lim .52-=+++-∞→x x x x。