微积分与数学分析微积分与数学分析

Calculus and Elementary AnalysiCalculus and Elementary Analysis (CEA)s (CEA)

主讲：　游雄助教：　魏敏

2007-2009

Instructor: Xiong You

Teaching Assistant: Min Wei2007.9-2009.1

NJAU mathematical analysisNJAU mathematical analysis

Mathematical AnalysisMathematical Analysis Calculus Calculus

and and Elementary AnalysisElementary Analysis

(CEA)(CEA)

Reference:

1. W. Rudin: Principles of mathematical analysis. McGraw Hill Inc., 1976.

Textbook: 1. East China Normal Uiv.: Mathematical analysis. 3rd edition. Higher Educational Press, 2002.(In Chinese)2. Finney, Weir and Giordano, Calculus 11th edition, Pearson Education,2005.3. M.H. Protter, , Basic Elementary Real AnalysisBasic Elementary Real Analysis, Springer, 2004, Springer, 2004..

Homework assignment: 20%Classroom performance: 10%

Final exam: 70%

Grade PolicyGrade Policy

§1.1 Real Numbers§1.2 Sets of Numbers, the Least- Upper-Bound Principle§1.3 Real Functions

§1.4 Composite Functions and Inverse Functions

NJAU mathematical analysisNJAU mathematical analysis

Chapter1 Chapter1 Real Numbers Real Numbers

and Real Functionsand Real Functions

§1.1 Real Numbers

a. Real numbers and their properties

b. Absolute values and inequalites

fractions (positive, negative, zero):

( finite or infinite cycling decimals)

: ( infinite

real

irrati decimal

rational

onal s)no

p

ncycling

q

a. Real numbers and their properties

[Recall] Rational and Irrational Numbers

Notation for Sets of numbers

R ----real numbers

Q ----rational numbers

Z ----integers

N ----natural numbers

0 1 2 (infinite noncyclin ). gna a a a

oror

￭ Write Real Numbers in form of decimals

1) Irrational

2) Rational

0 1 2 0 1 2. . ( 1)99 9n na a a a a a a a

0 1 2 1 2 1 2 (infinite cycling). r k ka a a a c c c c c c

￭ Comparison of Real Numbers

1) Definition 1 Given two nonnegative real numbers,

,.,. 210210 nn bbbbyaaaax

0 0,a b where

are nonnegative integers,

,i ia b are

integers with

0 , 9, 1,2, .i ia b i We say that

1) x=y , if , 0,1,2, ;i ia b i

2) x>y , if 0 0 ,a b or

0 0 , , 1, 2, , ,i ia b a b i k but

1 1.k ka b

For negative numbers,1) x=y, if -x= -y ;2) x>y , if -y= -x .

￭ Comparison of Real Numbers

Convention:

Nonnegative numbers > negative numbers

naaaax 210 .

nn aaaax 210 .

nnn xx10

1

Definition 2 Let be

a nonnegative number.

1) The rational number

is called the n th lower approximation of x ;

2) The rational number

is called the n th upper approximation of x .

0 1 2. ,nx a a a a

0 1 2

1. ,

10n n nx a a a a

0 1 2. .n nx a a a a

210 xxx

210 xxx

,

For a negative

we define

and

Note that for any real number x,

PROPOPSITION 1 For real numbers

0 1 2 0 1 2. , . ,x a a a y b b b

an int

if a

eger

nd only

, such that .there

i

f

is n n

x

n

y

x y

PROPERTIES OF REAL NUMBERS(1)

1. The set of real numbers is

closed under addition,

subtraction, multiplication and

division (with nonzero denominator),

that is, the sum, difference, product or

quotient (with nonzero denominator) of

two real numbers is a real number.

PROPERTIES OF REAL NUMBERS(2)

2. The set of real numbers is

ordered , that is, for two real numbers

a and b, exactly one of the following

relations is true:

a < b, a = b, or a > b .

PROPERTIES OF REAL NUMBERS(3)

3. The order of real numbers

is transitive , that is,

if

a<b and b<c,

then

a < c.

PROPERTIES OF REAL NUMBERS(4)

4. Real numbers has the Arc

himedes Property, that is, for an

y real number b and a>0, there is a natur

al number n

such that

b < na.

PROPERTIES OF REAL NUMBERS(5)

5. The real number set R is

dense, that is, between any two real

number a < b, there are other real

numbers, both rational and irrational.

PROPERTIES OF REAL

NUMBERS(6)

6. The real number set R are

one-to-one correspondent

to the set of points of the

real line. xx。。

Example 1

there is a rational number r, such that x < r < y.

Proof.

Let x and y be real numbers. Show that

Since x<y, there a nonnegative integer n, such

that .nn yx Let / 2.n nr x y Then r is

rational, and .n nx x r y y

Example 2 Let a and b be real numbers. Show that

If for any positive ε, a < b+ ε , then .a b

Proof. (Proof by contradiction) Suppose that a>b and

let ε = a-b. Then ε >0 and a = b + ε . This contradicts the

assumption. Therefore, it must be true that .a b

, 0,| |

, 0.

a aa

a a

a0

-a

b. Absolute values and

inequalities

从数轴上看的绝对值就是到原点的距离：

DEFINITION The absolute value |x| of a real number x is defined by

Some properties of absolute values1 | | | | 0 ; and | | 0 only when 0.a a a a .

2 . a a a - | | | | .

3. - < < ; | | , 0.a h h a h a h h a h h | |

.4. a b a b a b

5. | | | | | | .ab a b| |

6. , 0.| |

a ab

b b

QUESTION

Let a and b be real numbers.

If for any positive ε, |a – b|< ε , what

order is between a and b?

Some important inequalities(1):

2 20 2 ( , ).a b ab a b R

( 算术平均值 )

( 几何平均值 )

( 调和平均值 )

1 Mean value inequalities.

3 3 3 3 ( , , 0).a b c abc a b c

and the equality holds .

Some important inequalities(1)

1 2 ( , 0).a b ab a b

( 算术平均值 )

( 几何平均值 )

( 调和平均值 )

1 Mean value inequalities.

33 ( , , 0).a b c abc a b c

oror

2 222 ( , 0).

1 1 2

a bab a b a b

a b

In general,

⑵ 均值不等式 :

1 2for , , , ,na a a R

1 2

1

1 ,

nn

ii

a a aA a

n n

( 算术平均值 )1

1 21

,n n

nn i

i

G a a a a

1 2 1 1

1,

1 1 1 1 1 1n n

n i ii i

n nH

a a a n a a

,H G A Q

where

we have,

22 2 2

1 2 1 .

n

in i

aa a a

Qn n

Some important inequalities(2):

2 2 2 2 ( , , , ).ac bd a b c d a b c d R

( 算术平均值 )

( 调和平均值 )

2 Cauchy inequalities.

In general,

2 2

1 1 1

.n n n

i i i ii i i

a b a b

and the equality holds .

有平均值不等式 :等号当且仅当 时成立 .

⑶ Bernoulli 不等式 :

(1 ) 1 (1 ) 1 1 1n nx n x ).1( )1( xnxn n n

.1)1( nxx n

Some important inequalities(3):

3 Bernoulli inequalities.

,2,,0,1For nNnxx

Proof. (Proof by induction)

If 1 0,1 1, and 2,x x n N n

.1)1( nhh n

⑷ 利用二项展开式得到的不等式 :For 0,h

由二项展开式 ,!3

)2)(1(

!2

)1(1)1( 32 nn hh

nnnh

nnnhh

(1 )nh

More useful: since

then

any term, or the sum of any parts of r.h.s.

⑷ 利用二项展开式得到的不等式 :For 0,h

由二项展开式 ,!3

)2)(1(

!2

)1(1)1( 32 nn hh

nnnh

nnnhh

(1 )nh

Question: since

then

any term, or the sum of any parts of r.h.s.(right hand side).

If

Some important inequalities(4):

sin 1,x

sin .x x( 算术平均值 )

( 几何平均值 )

4 Other useful inequalities.

.tansin0

,2/0For

xxx

x

Homework Today P4. 3, 4, 6, 7

Class is over. See you next time!