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 :N ote

1)(,1

1)(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 )=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 -=4

22x

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 ===+=

2

sin 1

log

)(x

e

x x f x

a

-+

=Example 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 ∈++++=--0

11

1)( 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 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)

?

?

?>≤-=11

1)(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)()( 3) boundedness(有界性)

below

bounded )(x

e

x f =Example1

above

bounded )(x

e

x f -=Example2

below

and above from bounded sin )(x x f =E xample3

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 → E xample1

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 2

20

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

11

1

lim

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

except near 0)(,0)(lim )x f x f x f b x x

1

3124lim

42

3

+-+∞

→x x x x Example1

4

4

2

13124

lim

x x

x

x

x +-+

=∞

→ 0=

1

3322lim

2

2

++-∞

→n n n n Example2 2

213322lim

n

n

n n ++-

=∞

→ 3

2=

x

x x x 7812lim

2

3

++∞

→E xample3 2

3

781

2lim

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

1

101

1 ,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

22

lim

)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

4

31

43lim

)1.22

2

=

++∞

→n n n n 5

1)

2(5

)2(5lim

)21

1

=

-+-+++∞

→n n n n n

343

13

1121

211lim

)3=+

+++++

∞

→n

n

n 1)1231(lim )4222=-+++∞→n

n n n n 1))

1(13

212

11

(

lim )5=++

+?+?∞

→n n n 2

1)1(lim )6=

-

+∞

→n n n n

∞=---→4

43lim

)1.32

2

2

x x x x 2

3

30

3)(lim

)2x h

x

h x h =-+→

34

3153lim

)32

2=++++∞

→x x x x x 50

30

20

50

30

20

5

3

2)

15()

23()

32(lim

)4?=

+++-∞

→x x x x

2)12)(11(lim )52

=-

+

∞

→x

x

x 07

24132lim

)65

3

=++++∞

→x x x x x

4

21

13lim

)72

1

-=-+-

-→x x

x x 1)1311(

lim )83

1

-=---→x

x

x

3

21

1lim

)93

1

=

--→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

2

3

)

3(3lim

)1.4x x x x ∞=++∞

→4

32lim

)23

x x x

∞=+-∞

→)325(lim )32

x x x

1)2544(lim .52

-=+++-∞

→x x x x