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微积分与数学分析 Calculus and Elementary Analysis (CEA) 主讲: 助教: 游雄 魏敏 2007-2009 NJAU mathematical analysis Mathematical Analysis Calculus and Elementary Analysis (CEA) Instructor: Xiong You Teaching Assistant: Min Wei 2007.9-2009.1 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 Analysis, Springer, 2004. Reference: 1. W. Rudin: Principles of mathematical analysis. McGraw Hill Inc., 1976. Grade Policy Homework assignment: 20% Classroom performance: 10% Final exam: 70% NJAU mathematical analysis Chapter1 Real Numbers and Real Functions §1.1 §1.2 Least§1.3 §1.4 Real Numbers Sets of Numbers, the RealUpper-Bound Functions Principle Composite Functions and Inverse Functions §1.1 Real Numbers a. Real numbers and their properties b. Absolute values and inequalites a. Real numbers and their properties [Recall] Rational and Irrational Numbers p fractions (positive, negative, zero) rational: q real ( finite or infinite cycling decimals) irrational: ( infinite noncycling decimals) Notation for Sets of numbers R ----real numbers Q ----rational numbers Z ----integers N ----natural numbers ■ Write Real Numbers in form of decimals 1) Irrational a0 .a1a2 an (infinite noncycling) 2) Rational a0 .a1a2 ar c1c2 ck c1c2 ck (infinite cycling) or a0 .a1a2 an a0 .a1a2 (an 1)99 9 ■ Comparison of Real Numbers 1) Definition 1 Given two nonnegative real numbers, x a0 .a1a2 an , y b0 .b1b2 bn , where a0 , b0 are nonnegative integers, ai , bi are integers with 0 ai , bi 9, i 1, 2, 1) x=y , if ai bi , i 0,1, 2, . We say that ; 2) x>y , if a0 b0 , or a0 b0 , ai bi , i 1, 2, , k , but ak 1 bk 1. ■ Comparison of Real Numbers Convention: Nonnegative numbers > negative numbers For negative numbers, 1) x=y, if -x= -y ; 2) x>y , if -y= -x . Definition 2 Let x a0 .a1a2 an be a nonnegative number. 1) The rational number x n a0 .a1a2 an is called the n th lower approximation of x ; 2) The rational number xn xn 1 n 10 is called the n th upper approximation of x . For a negative we define and x a0 .a1a2 xn a0 .a1a2 an , 1 an , n 10 xn a0 .a1a2 an . Note that for any real number x, x0 x1 x2 , x0 x1 x2 PROPOPSITION 1 For real numbers x a0 .a1a2 , y b0 .b1b2 , x y if and only if there is an integer n , such that xn yn . 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 Archimedes Property, that is, for any real number b and a>0, there is a natural 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. x 。 Example 1 Let x and y be real numbers. Show that there is a rational number r, such that x < r < y. Proof. Since x<y, there a nonnegative integer n, such that x n yn . Let rational, and r x n yn / 2. x xn r yn y. Then r is 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. b. Absolute values and inequalities DEFINITION The absolute value |x| of a real number x is defined by a , a 0, | a | a , a 0. 从数轴上看的绝对值就是到原点的距离: -a 0 a Some properties of absolute values 1. | a | | a | 0 ; and | a | 0 only when a 0. 2 . - | a| a | a| . 3. | a| h -h < a < h ; | a | h h a h , h 0. 4. a b a b a b . 5. | ab || a || b |. 6. a |a| , b 0. 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): 1 Mean value inequalities. 0 a b 2ab (a, b R). 2 2 and the equality holds . (算术平均值) (几何平均值) a b c 3abc (a, b, c 0). 3 3 3 (调和平均值) Some important inequalities(1) 1 Mean value inequalities. 1 a b 2 ab (a, b 0). (算术平均值) a b c 3 abc (a, b, c 0). 3 or (几何平均值) ab 2 ab a 2 b 2 (a, b 0). 1 1 2 (调和平均值) a b 2 for a , a , , a R , we have, In general, 1 2 n H G A Q, n where H ⑵ 均值不等式: 1 1 a1 a2 G 1 an 1 n 1 1 n i 1 ai n n , 1 i(算术平均值) 1 ai 1 n n a1a2 a1 a2 A n an an n ai , i 1 1 n ai , n i 1 n Q a a2 n 2 1 2 an 2 a i 1 n 2 i . Some important inequalities(2): 2 Cauchy inequalities. ac bd a2 b2 c2 d 2 (a, b, c, d R). (算术平均值) In general, n a b i 1 i i and the equality holds n n a b 2 i 1 i i 1 i 2 . (调和平均值) . Some important inequalities(3): 有平均值不等式: 3 Bernoulli inequalities. 等号当且仅当 ⑶ Bernoulli 不等式: For x 1, x 0, n N , n 2, 时成立. (1 h) n 1 nh. Proof. (Proof by induction) If x 1 0,1 x 1, n N and n 2, (1 x) n 1 (1 x) 1 1 1 n n n n (1 x ) n n (1 x ). (1 x) 1 nx. n For h 0, since ⑷ 利用二项展开式得到的不等式: n(n 1) 2 n(n 1)( n 2) 3 n (1 h) 1 nh 由二项展开式 h h hn , 2! 3! More useful: then (1 h) any term, or the sum of any parts of r.h.s. n For h 0, since ⑷ 利用二项展开式得到的不等式: n(n 1) 2 n(n 1)( n 2) 3 n (1 h) 1 nh 由二项展开式 h h hn , 2! 3! Question: If then (1 h) any term, or the sum of any parts of r.h.s. n (right hand side). Some important inequalities(4): 4 Other useful inequalities. sin x 1, (算术平均值) sin x x . For 0 x / 2, 0 sin x x tan x. (几何平均值) Homework Today P4. 3, 4, 6, 7 Class is over. See you next time!