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 :Note

1)(,1

1)(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 functions

f (x ）=c

2) power functions

0,)(≠=a x x f a

3) exponential functions

1,0,)(≠>=a a a x f x domain: R range: ),0(∞

4) logarithmic functions

1,0,log )(≠>=a a x x f a domain: ),0(∞ range ： R

5） trigonometric functions

f (x )=sin x f (x ）=cos x f （x )=tan x f （x ）=cot x f (x )=sec x f （x ）=csc x

Given 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 x

x x g x f g x f g b -===2)())(())(()

]4,0[:0

2,0domain 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[:0

22,02-????≥--≥-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)(2

x h g f x F x x f x x g x x h ===+=

2sin 1

log )(x e x x f x a -+=E xample is an elementary function 。

1）Polynomial （多项式) Functions

R 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

In particular （特别地)，

The leading coefficient ?≠.00a constant function

The leading coefficient ?≠.01a linear function

The leading coefficient ?≠.02a quadratic(二次） function

The leading coefficient ?≠.03a cubic(三次） function

2）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 Functions

4。Piecewise Defined Functions （分段函数)

???>≤-=111)(x if x

x 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 xample1

above bounded )(x e x f -=E xamp le2

below and above from bounded sin )(x x f =Example3

4） periodicity (周期性)

Example f （x )=sin x

Chapter 2 Limits and Continuity

1.Definition We write L x f a

x =→)(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 Laws

Suppose that c is a constant and the limits )(lim and )(lim x g x f a

x a x →→exist 。 Then )(lim )(lim )]()([lim )1x g x f x g x f a

x a x a x →→→±=± )(lim )(lim )]()([lim )2x g x f x g x f a

x a x a x →→→?= 0)(lim )(lim )(lim )()(lim )3≠=→→→→x g if x g x f x g x f a x a

x a x a x Note From 2)， we have

)(lim )(lim x f c x cf a

x a x →→= integer. positive a is ,)](lim [)]([lim n x f x f n a

x n a x →→= 3。

1）

2)

Note

4.One —Sided Limits

1）left-hand limit

Definition We write L x f a

x =-→)(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 L x f a

x =+→)(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 a

x a x a x +-→→→==?= ||lim Find 0

x x → Example1 Solution

x

x x ||lim

Find 0→ Example2 Solution

6。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 220

0lim x x x ?=→ is an infinitesimal as .0→x Example2 x

x 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 x

Note Example 01sin lim 0=→x

x x 3）Definition ?±∞=?

→)(lim x f x We say f （x ） is an infinity as ??→ where ,x is some number or .∞± Example1 1

111lim 1-?∞=-+→x x x is an infinity as .1+→x Example2 22lim x x x ?∞=∞

→ is an infinity as .∞→x 4)Theorem

0)

(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 42

3+-+∞→x x x x E xample1 44213124lim x

x x x x +-+=∞→ 0= 13322lim 22++-∞→n n n n E xample2 2213322lim n

n n n ++-=∞→ 32= x x x x 7812lim 23++∞→E xample3 237812lim x x x x ++=∞→ ∞=

Note ????

?????>∞<==++++++-----∞→m n if m n if m

n 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(31

22lim )1.12==?=-++∞→b a n bn an n )1(),1(1)1(lim )22-==?=--+∞→b a b ax x

x x )2(),2(21

lim )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 ∞=---→4

43lim )1.3222x x x x 23

303)(lim )2x h x h x h =-+→ 343153lim )322=++++∞→x x x x x 50

30

20503020532)15()23()32(lim )4?=+++-∞→x x x x 2)12)(11(lim )52=-+∞→x

x x 0724132lim )653=++++∞→x x x x x 42113lim )721

-=-+--→x x x x 1)1311(lim )83

1-=---→x x x 3

211

lim )931=--→x x x 61)31)(21)(1(lim )100=-+++→x x x x x

2

1))1)(2((lim )11=--++∞→x x x x ∞=-+→2

23)3(3lim )1.4x x x x ∞=++∞→432lim )23x x x ∞=+-∞

→)325(lim )32x x x 1)2544(lim .52-=+++-∞

→x x x x