微积分与数学分析 calculus and elementary analysis (cea) 主讲: 游雄 助教: 魏敏...
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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..
§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
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